hi, im wondering about the following, my notes say that in $\mathbb{D}$ with the poincare metric $p(z) = \frac{2}{1 - |z|^2}$ and corresponding distance defined in terms of an infimum of path lengths, if $-1 < x < y < 1$ then the shortest path from $x$ to $y$ is the 'simple arc' connecting them, presumably this just means the straight line, but I found that as long as $\gamma : [0,1] \rightarrow [x,y]$ is piecewise $C^1$ and (nonstrictly) increasing, it is also a shortest path

is there some sense in which any such $\gamma$ is equivalent to the straight line path $[0,1] \rightarrow [x,y]$?

here I mean $\gamma([0,1]) = [x,y]$

ideally is there a way of saying any such $\gamma$ is 'just' a 'reparameterization' of the straight line path

@porridgemathematics Length of a curve does not depend on parametrization, ever.

yes im aware

but both $\gamma$ and the straight line path are paths, that are in some sense representatives of the same 'curve'?

Only if you want a geodesic, defined as a parametrized curve with parallel tangent vector, do you need a constant-speed path.

You can arclength parametrize the curve, at which point it'll be a straight-line path.

Ted beats me to it.

ah okay, I am missing the definition of a 'curve' here, is 'curve' really a one-dimensional manifold with boundary?

A path is a parametric curve. Look up reparametrization.

No, we're talking about smooth maps from an interval.

Piecewise $C^1$.

OK :)

Look in the first section of my diff geo text, porridge.

okay, so that is what I was asking, $\gamma$ and the straight line paths are just parameterizations of the same 'curve' because there is some $t : [0,1] \rightarrow [0,1]$ that is increasing, for which $\gamma \circ t $ is the straight line path?

and $t$ has some smoothness? Perhaps it is piecewise $C^1$ as well?

chat.stackexchange.com/transcript/message/58545300#58545300 wait so Americans arent thought calculus in high school? (i wanna know if i failed trying this replying option)

It doesn’t have to be the same interval. Usually $C^1$ with nonzero derivative.

well sure but it can be, okay I am glad you said that, but in this case it can't be $C^1$ with nonzero derivative because then we would be saying a piecewise $C^1$ curve is really $C^1$ , when that needn't be the case

adil there is no national system of education in the united states. there are approximately 15,000 school districts that address 50 state standards in varying ways.

in 1920 when robjohn was a boy it would not surprise me that a public high school did not teach calculus. these days it is more common.

Most every high school offers calculus now. Not so when robjohn and I went to high school.

it was even less common in 1900 when ted was a boy.

I'm pretty sure calculus wasn't taught 50 or so years back in India. My father learnt it in college when he took economics.

my parents did not take calculus in high school. only my dad went to college, it was normal not to have taken calculus then.

It's taught now in high school but taught completely wrong.

his high school was also racially segregated in violation of brown v board of education. the 1960s were not a great time.

so my question was in this case, when $\gamma$ is piecewise $C^1$, not necessarily $C^1$, the notion of reparameterization is not 'as per usual' where the map $t : [a,b] \rightarrow [a',b']$ is $C^1$ with nonzero derivative, so is there a standard class $t$ would be in this case?

wait which specific part @balarka?

@AdilMohammed :58545300 instead of the link...

would it just be piecewise $C^1$ with nonzero derivative, rather than $C^1$?

@leslietownes Jim Crow?

Maybe that's before 60s

@AdilMohammed I don't understand your question. Which specific part of what?

it was decided in the memphis public school system that people who hadn't been in integrated classrooms would not be able to handle them. so they integrated kindergarten, then first grade, then second grade. the supreme court decision said it was unconstitutional to segregate and there was no basis for this slicing and dicing of various levels of segregation.

@Wolgwang oh and you click the arrow, makes sense thanks

i think they were worried about what would happen at prom.

@BalarkaSen this one "It's taught now in high school but taught completely wrong."

Oh, every part. Most 12th graders do not understand calculus, they just know integration rules-of-thumb.

@porridge You just want the arclength integral to make sense.

the AP curriculum in the united states, which is fairly uniform, is the same way. lots of rule-based memorization and no understanding.

:'-(

Well, there are some very good teachers, leslie, but on average …

i had a very good calculus instructor but he was working within the AP curriculum which i think was suboptimal.

yeah I realize that the notion of reparameterization is so that the length of a 'curve' is independent of parameterization.. but that also makes a 'curve' an equivalence class of paths with the same image, where the equivalence relation is the reparameterization, and I was asking what the equivalence relation needs to be when you start thinking about piecewise $C^1$ rather than $C^1$ curves, because it can't be exactly the same

@leslietownes tru... my teacher said "just by heart differentiation of log and e^x for time being and you crack your head later after class"

I was lucky to have a been taught in high school by a math PhD, which is pretty rare.

but I understand this isn't a big issue

Minimal emphasis on word problem skills, which is the whole point for me.

he lent me a bunch of college math books when i was in high school. he had a master's degree in math which was very rare.

i was obsessed with his book on numerical analysis. i implemented most of the algorithms on my calculator and even found a typo in the book.

@porridge Just think it through for something like one corner.

I sort of understood calculus in one variable because by father gave his copies of Piskunov's classics to me, which he used in college.

For a long time I had no understanding of multivariable calculus. The credit there goes to Ted

In the 1960s, some US states still had laws banning interracial marriage. en.wikipedia.org/wiki/Loving_v._Virginia

it's fairly common for math instructors in the US not to have math degrees. they more commonly have degrees in education. they tend not to know what math is about, because the textbook world is so different from the real world.

@TedShifrin I would just use two smooth reparameterizations (with nonzero derivative on each subinterval)

math is not a popular subject. my wife's college graduating class had something like 3000 students in it. we had 45.

@leslietownes that reminds me i too used to make a collection of mistakes in a google doc that our board made while printing the tb

And there is little to no incentive for math students to go into teaching in high schools. Low pay, bad environment, mind numbing work...

im guessing that generalizes to the general case, where you just take a refinement of the partition corresponding to each piecewise $C^1$ path, and do the same thing

But mathematicians are often terrible teachers, so knowing a lot isn’t always an advantage. Plus, there’s a tendency to forget that one’s students aren’t little versions of self.

balarka that is a good point. a lot of my friends in college wanted to be high school instructors and have since quit.

@BalarkaSen hmm yeah

there is way too much emphasis in math on encouraging prodigies and identifying talent at a young age, and not enough emphasis on communicating to the actual students you are likely to have in your classroom.

it's easier to imagine everyone as mini mathematicians.

@porridgemathematics Sounds reasonable to me. Of course, you can smoothly parametrize if you allow vanishing derivatives. Do you want to allow that?

hm, I don't see how you can, take $\gamma(t) = t$ from $[0,1] \rightarrow [0,1]$ and a piecewise continuous path from $0 $ to $1$ (increasing) with one corner, if there was a smooth reparameterziation $u$ for which $\gamma \circ u$ was the piecewise continuous path with one corner, then the piecewise continuous path would have a well-defined derivative at the corner..

Don’t think about reparametrizing. Can you parametrize the path $(0,1)\to (0,0)\to (1,0)$ smoothly.

yes, using a bump function

@leslietownes It is actually very easy to imagine teaching kids math but in execution it seems 10 fold less glamorous than it sounds. Mostly not because it's hard work although it takes dedication in part of the teacher, because kids are often high energy (they're always trying to minimize entropy!) and your task as teacher is to channel that energy into problem solving and critical thinking or whatever.

But the sad part is it's soul-consuming to teach them the same dry stuff out of the normalized textbooks, be involved in department bureaucracy and then go home after a 9-5 with minimal pay. Of course you're going to stop caring after some point.

but I thought we were literally just talking about reparameterziations?

I just wanted to make a side point. For reparametrization you want diffeos on each piece, yes.

even though the arc length integral would be the same in this case, we would need to change our notion of two paths tracing the same curve

Well, the trace is just a matter of the sets.

I mean yeah, but I guess in my head I'm also preserving orientation

@BalarkaSen "department beurocracy..." *makes raspberry blowing noise

Yes, although for length that is unnecessary.

right

but we can't backtrack to preserve length , right?

Right.

even though we can preserve the image that way

okay, just checking

You need bijections.

okay, so i guess the most general version of a reparameterization is like a bijection $[a,b] \rightarrow [a',b']$ that is absolutely continuous

i feel like that would preserve length

and would be differentiable a.e.

so im guessing the chain rule thingy would work

i was pretty well evaluated on average and even won a teaching award but i could not do high school. people are coming into the class with almost nothing. i need something.

I don’t want to think in that generality unless it’s GMT, but i suppose …

whats GMT?

oh, geometric measure theory?

Yeah

@BalarkaSen i think its bs that our curriculum (indian) invests so much time in changing the question paper every two years, but forgets to prepare students for basically everything... best example would be 11th grade phys/chem... believe me my physics/chem teacher shouldn't be the one introducing me to calclus.

@porridgemathematics Also known as differential geometry in batshit low regularity

so 'weirder' DG

sounds like fun

When my ability to motivate students to work their butts off diminished (I would argue not because I changed), I decided to retire. Still motivated most, but not nearly all.

my wife has what i would think of as high school issues in college now. some of her students are homeless. i couldn't deal with that. i can't get you to care about math if that's what you're going through.

@leslietownes yeah, big problem in india

Indeed

most students in rural schools are from extremely low income families

i studied in a couple of such schools

tough issues.

albany has a few homeless students. turned up when doing an 'equity' analysis.

people are also dealing with covid deaths without a lot of help.

@AdilMohammed (and whoever interested) Here is a serious take on Indian (math) education by M S Raghunathan, a famous mathematician (to the advanced crowd: he proved arithmetic groups have finitely generated cohomology using Morse theory):

people worry about marginal differences when there are people with real problems.

First 15 minutes or so is history, and then he moves on

is my question clear? what to do first? :(

ms raghunathan needs no introduction. if i've heard of you and i don't work in your field, you're famous. :)

:)

He's a great inspiration for many of us who has had the chance to interact with him.

sorry for my slow typing skills, continuing my rant in the education industry in India, (i believe there are some US similarities too) so we enter 11th-12th not having being taught what i consider pre-requisite information in phys/chem like calculus and we are taught calc only in maths towards the end .

and we leave school not having what colleges consider pre-requisite information to study in college. and we leave college not having what workplaces considers pre-requisite information to work (i asked a couple of engineers in my family including my bro apparently things like primavera and other software which are needed in workplaces are not taught in colleges, so a couple of months is spent in teaching those skills which is why employers are reluctant to hire new grads)

@SAJW i presume you need $P[a=H] = P[a=T,b=H]=P[a=T,b=T,c=H]$.

yes, that is often how it works in the US, too.

i was given no education in physics until my final year.

@leslietownes hmmmm... crap education sector we both got

it would have helped me understand calculus.

we learned calculus fairly early on. lots of gaps in other aspects though.

@leslietownes yessss, for us only 11th and 12th real physics is being thought (ie involving math, like not just qualitative but quantitive)

i did have a very good physics instructor in high school. if i'd had him for 9th grade, maybe i'd be a physicist.

but i was motivated by all the mysterious symbols in my dad's schaum outline for financial mathematics...

@copper.hat ah, indeed

i was motivated by listening to my dad taking tuitions at home.... early exposure is very important in my opinion in everything

i still remember staring at heron's formula and thinking "the hell is that??"

pick's theorem was an early introduction to math for me.

also anyone got any great video suggestion for introduction to log and e?

i think the quadratic formula was for me (primary school). my dad stayed up all night trying to remember the proof. i think that left its mark on me that someone would care enough about a formula

@copper.hat Hah! The same thing got me into math

and now i'm hungry for pizza. thank you.

I posted my answer, which I thing it makes sense than theirs! Oh, hey mentioned that at least one pizza! My bad. I will delete my answer now.

I ordered pizza 3 days ago!

At some point me and my dad would sit and talk for hours on end trying to decipher Galois theory texts to understand what's up with quintics.

Yeah, things escalated quickly after quadratics

i have a work lunch and a work dinner so all of my food is free today. one of the meals might be pizza.

ooooh someones lucking tonight

**lucky

@BalarkaSen wow, that is cool.

i'm thinking pizza for lunch and sushi for dinner.

One of the books we were reading was by a chemist, R B King, and @Ted would later end up telling me he's from UGA. It's a surprisingly small world.

@BalarkaSen I didn’t realize your dad is such a mathy!

@BalarkaSen in my case my dad was a teacher upto 10th grade only, so i remember "teaching" him differentiation, integration, functions and all those things late night

when the uae education ministry made every math teacher do a test for renewal of teachers license

That's cool!

lol not as much as talking galoise theorem thou

*talking about galoise theorem with your father

my dad taught me to type. not nothing. everyone in his family were writers.

It would have been cool if we understood Galois' theorem tho

@leslietownes Like on a typewriter?

yes, on my grandfather's typewriter.

it was an old machine. somehow we could still find ribbons for it.

carriage return and line feed are separate operations. a lot of kids don't know that.

carriage return is when you ram the piston back in right? that doesn't automatically put a new line?

yeah. you can slap the carriage back to the home position but you need to move the lever to advance it to a new line.

Wow!

My life was a lie

i remember the introduction of the ibm golfball typewriters

so cool to be able to change font easily

those are really cool. not old enough to remember their existence, but i saw a video about them a bit ago

they had some of them in the copy room at evans hall.

my dad's office used to let me use their telex machine to send non confidential messages to customers.

early internet :-)

not quite

when i bought a fax modem in 1993 the first thing i did was fax a page of black to my dad's office.

i added a comment on the order of "here you go."

i was annoying even then.

lol

i was not an easy child to raise and now my daughter is paying me back.

to deal with an identity theft issue for my then 17 yo son i had to fax some stuff to the credit agencies. we're talking a few months ago.

my wife's had her identity stolen twice. her SSN has needed changing, and then people act like she's the fraudster for having all of these identities.

wow. that's awful.

@copper.hat 1/9, 1/8, 1/7 works, but that was simply a guess and I think it works for other values too.

she changed her last name so it matches mine. i told her that otherwise people might think our child was illegitimate, and that would be shameful.

@SAJW i see no reason why the answer should be unique.

i don't either.

@BalarkaSen Well, you could do the carriage return & press the linefeed lever while the carriage was returning. If you watch people typing on old movies, it looks like a single smooth motion. (I learned on my grandmother's typewriter, which was from the 1950s or earlier).

there was that funny tape you added in to correct mistakes

Like anti carbon paper.

you definitely do it as a smooth motion if you have the option of doing so.

as kids we loved carbon paper

we tried using it for lines (school punishment)

oh yeah the tape i have seen

and i have definitely messed around with carbon paper

school stuff was copied on a mimeograph

remember that fluid? the duplication fluid that made things purple?

then again, we had inkwells and used quills (the plastic/wooden handle variety)

wow.

yes, it smudged and made stuff hard to read

the resolution was not great.

i still use a fountain pen from time to time

that seems like an affectation to me.

@TedShifrin did you ever do flipped classroom stuff?

i like the feel of a fountain pen.

it is an old pen from an uncle.

i think it looks nicer when writing condolences, etc

it's not an affectation if it's from an uncle.

i have my grandfather's pocketwatch. a pocketwatch is ludicrous but not if it's your grandfather's pocketwatch.

:-). it was a little luxury back then.

i mentioned that my son has a 1965 omega seamaster from another uncle.

of course he wears his fossil modern watch

on sale in vacaville

Modern thermoprinter paper can be fun for kids to play with. It's the standard these days for cash registers, since it works without an ink ribbon. Heat from friction can make it go dark, so you can do stuff like coin rubbings with it.

oooh

i think there are many older things that kids would love to play with

maybe not mercury like we had

my memory don't work so good since i drank my thermometer.

at least i think that is what it was called :-)

beat me to it

i'm on a call with about ten lawyers and some of them are yelling at one another.

i'm not paid enough to yell at anybody.

I don't have a working fountain pen, but I do have a dip pen & ink.

@copper.hat Gallium is a safer alternative.

A little more fun, too, since you can make it solid easily.

i hate having to do this. i've made people cry in depositions and when they start crying you're supposed to go to the best questions. you feel horrible while you're doing it.

I used to use it on Christmas cards... when I still gave Christmas cards. ;)

i made my daughter cry when we worked on her essays. i was oblivious of course.

@leslietownes is your bonus proportional to the number of tears you cause them to shed?

sometimes it feels like that.

i don't mind it when they deserve it

god, people are yelling at each other. i hate this.

@anakhro Gallium dissolves aluminium oxide. Aluminium is actually quite reactive, it even reacts with water to produce hydrogen (although not as vigorously as sodium does). But that normally doesn't happen because it rapidly forms a protective oxide layer.

i wish i had some vague intuition for chemistry

leslie's meeting: youtube.com/watch?v=2Xug_o2iXPo

chemistry is organized chaos.

Guess which one is leslie

hahahaha

i hate yelling and there is a lot of toxic masculinity on this call, someone on our side is being dismissed and i don't know what to do about it.

ugh.

@PM2Ring The classic popcan experiment is fun.

pretty mild compared to many meetings i have been in

@copper.hat You might like Pauling's chemistry book. it's nice.

ok, i just said something offensive. but it made people laugh.

let's laugh instead of being toxic.

@anakhro which one?

@copper.hat General Chemistry. It's available as a Dover book and is cheap and loved.

@copper.hat so for n rigged coins it has to be $P(x_1=H)=P(x_1=T & x_2=H)=...=P(x_1=T& x_2=T & ... x_n=H$ ?

people are actually being normal now. my profanity may have alleviated this nonsense.

@anakhro thanks. have just ordered it.

Quick to shop. :P

i told the other side to go f themselves. people laughed because it was what they were all thinking.

@SAJW yes. i think that is was i wrote above?

leslie fought toxic with toxic and now it is not toxic

i said, if that's your position you can go f yourself. laughter. the call became normal after that.

profanity can be healing.

i did stand on top of the meeting table to make a point once. not sure what i was thinking.

i am fairly certain there was shouting involved.

@anakhro I've never done it myself, I've only seen videos.

now that performance would require slow, carefully computed motions to avoid age related injuries.

@copper.hat I think getting the maximum chance for the first coin is a better task, or at least proving there are (i think) infinite solutions.

@SAJW the implicit function theorem might help.

I've been doing more grid walks using residues of primes. This time, I'm using mod 30, which has 8 coprime residues, (1, 7, 11, 13, 17, 19, 23, 29), which we can map to the 8 principal orthogonal & diagonal directions. There's no obvious mapping, so my code lets you assign any permutation of the residues to the directions, but there's a shortcut if you want to use a symmetrical permutation, i.e., each residue $r$ is assigned to the opposite direction to $30-r$.

reminds me of the map of europe+asia a bit, tbh :p

It's kind of strange that it's mostly in the right half-plane. That's not unusual. Some are even mostly confined to one quadrant.

I guess something complicated is happening in the interaction between the orthogonal & diagonal directions. I forgot to mention, that image uses the primes <5,000,000

sagecell.sagemath.org/…

^ There's the script.

I didn't even knew really what imlicite means (I kinda avoided that word for my whole life haha)

implicite means there is something, explicite tells us what exactly?

This one is upto 750,000, with the default permutation of residues.

It's basically the western side of the previous image.

@SAJW implicit in this context means that it gives the existence of a suitable function rather than an explicit formula.

@copper.hat yes, I watch a video about this theorem right now^^

just saying never really understood both terms

@copper.hat is it trivial to show that in order for $x_1$ to be maximal $x_n$ has to be exactly $1$?

oh, i mean $P(x_1=H)$ and $P(x_n=H)$

those mathstudents seem to understand either everything or are afraid to ask lol

I don't understand why the implicit function theroem gives us the formula for the deriviative of the implicit function.

Hey

mathb.in/59461

Could someone please see if what I'm doing is correct?

The extremely weird looking numbers concern me

@SAJW it doesn't. it shows that there is some differentiable function $g$ such that $f(x,g(x)) = 0$ and if you differentiate this with respect to $x$ you get a formula that gives the derivative of the implicit function.

ok, so for today I'm finished, will continue my mental excersising tomorrow :D

@porridgemathematics Hi, have you seen my message?

which one?

did you email it to me?

@A-LevelStudent

Good evening everyone!

does anyone know how to show that the group of conformal automorphisms of the unit disk acts transitively on the set of (euclidean) circles in the unit disk?

I read multiple times people saying "proof it", "I need to proof the following...". It shocks me, I would always use "prove", but as I am a non native speaker I was wondering whether it could be said or not.

That is grammatically incorrect.

Ok thank you!

@porridgemathematics No, sorry, I mean the message in our chatroom. Would you prefer me to email you?

It's still pretty clear what is intended, but using "proof" as a verb is incorrect here. I think it is correct under other contexts - you can fireproof or waterproof something for example, but I don't know if that extends to broader applications where "proof" itself can be used as a verb.

@A-LevelStudent oh, im not sure if I did, i'll check again and respond there, no thats ok

@hyper-neutrino Of course it it. What shocked me is that it happened with several users (though it is quite rare), I agree with you; one can also proofread. Thanks a lot for answering!

Grammar is quite tricky, and I respect anyone who isn't a native speaker trying to learn a new language, especially one as inconsistent and irritating as English :P No worries though.

english grammar and spelling is a disaster.

as my daughter said while i was videoing her yesterday: "that's enough taking me a picture."

There are plenty of native speakers who make gross grammatical errors. I would not worry about it.