I have $g = 1/u$, if I differentiate that wrt to t I get $\frac{t}{u}$

now I have a $t$ I don't know how to handle

but I didn't recall having this problem when I went through this on paper yesterday

when i write u = blah my usual next calculation, putting aside whatever algebra i might do later, is computing du = blah

if you insist on writing g(t) in terms of u first, when you differentte that you get g'(t) dt = [something] du

it's important to keep track of both left and right hand sides of whatever you're differentiating

$\frac{du}{dt} = -\frac{1}{(g(t))^2} \cdot \mathbf{g'(t)}$

doh chain rul

it ain't the rope rule

or the twine rule

ahh, and on my piece of paper I had undelined the chain rule because of our last conversation

huhuhu

I need to burn this into my brain somehow

look at mathjax not aligning the - with the bar of the fraction there. do you see that too or is it a chrome thing

I should stop mentally asking ted, and just think how would leslie see this :P

it must not like the way i did the superscript in the denominator

looks fine to me?

that looks fine to me but the one i wrote has the minus sign about one pixel below the fraction bar

Make SIMPLEST possible substitution!

there si a simpler one?

if you were listening to your inner ted you might've gone with u = g(t)

if I chain it properly u=1/gt worked pretty well on paper the other day

oh

huh

let me try that

lots of substitutions will work, and you can even undo a goofy substitution with other substitutions, but it's a question of how much algebra you want to do. simpler substitutions minimize that

(in a few minutes, once I have this typed out, I want to preserve my stupid way)

Preserve by pickling?

preserve by typing into a document I hope I will reference in the future (as opposed to the reams of paper I throw away)

earth day every day.

I saved the paper from land fill (in my defence)

remember, if your screen is dark text on a light background you want to use lots of big beefy letters to darken those pixels and save energy. but it's the opposite if you plan on printing.

I'm not at that level of caring for the planet

i used to do that. i'd go to the recycling bin in the computer lab and grab sheets out of it. scribble on that, tex my work when i had it where i wanted it, then back to the recycling bin it went. i should have kept that paper. i could have sold images of it as NFTs

lol, we moved building a few years ago, and they had all this letter heads they were going to throw away because it had the old address on, so I stashed em in my locker and have been scawling maths on ever since :P

(well maths and drawings of pigeons)

(actually more pigeons than maths)

i'm 3 days after my second shot and my arm still hurts. but my joints have stopped aching. it never got bad for me.

the first shot, weirdly, put me out of commission for about a day.

I'm actually 3 days after my second one, stopped noticing it the day after.

i read that apparently if you've been exposed to it, which i have reason to think i was, the first shot can be the whopper that the second shot is for a lot of people.

my wife felt almost nothing from both shots.

see, women are tougher

Yeah, I've heard that too

(poorly named IMO)

"Pfizer presents COVID arm"?

good branding opportunity

Apparently it happens with Moderna. Lol.

My friend also bought me one of these

etsy.com/listing/1000769651/vaccinated-sweatshirt-pfizer-alumni

have math textbooks started coming out with made up 'word problems' that use vaccination or virus transmission hypotheticals in them

if not, brace yourself, they're coming

I'm so glad I resigned from teaching

join the club! who else can we convert?

I'm glad I gave my students the experience of pulling data directly from Johns Hopkins' COVID data, but they can keep it to themselves and get themselves employed

i like teaching my daughter to recognize birds. if i could get a job doing that maybe i'd go back

The problem with teaching is that it takes an extraordinary amount of effort to do well, especially online, and as an adjunct, there is no incentive to putting all of that work in other than making connections with your students. I appreciate that aspect of it, but getting paid only $1800 over an entire semester? Nah.

it is a pity that people who actually care get so little support when mediocre is order of the day.

not a comment on you obv, but caring is just as valuable as nous.

Yeah, my department chair and I complained about that in private.

i knew one person who made a fairly strong go of adjuncting. he stacked up several near-identical classes at different colleges so the money made a little more sense for the effort he was putting in, but he was always driving to and from different colleges.

I get it.

the problem is that teaching is not va;lued in the usa

andrew: as an aesthetic matter i'd just write g'(t) in the first integral to get rid of the partials, and drop that mathrm affectation you seem to have acquired from unsavory influences.

I would say it's not just teaching. I think it's also this attitude that anything that anyone would ever need to know in the workplace doesn't involve anything more than high-school algebra.

i have a niece & a friend's daughter in ireland, both smart, capable personable kids who will struggle to get into teacher training college because of competition.

Heck, I remember back in my actuarial position that I felt excited using modular arithmetic in Excel.

my dad taught me 'casting out the nines'

i thought i had learned an incredible accounting secret

it is pretty cool :-)

i guess i can't really expense my starmuck purchases since i am working from home. not that the irs will look...

my parents never knew anything more than high school algebra. it was all they ever needed to know in the workplace.

that would have been the level of mine as well, i think, but it was fairly normal for the time.

is starmuck simply the closest thing? are the better cafes not open?

i think part of it is the rote teaching style, learn this, do exam, forget it.

I've been very lucky. I only taught for four semesters, and I occasionally hear back from students who ask for copies of lecture notes I gave them back when they took my class.

there are but since its hard to arraneg to meet folks for a coffee/tea i get starmucks

Turns out that some of them figure out that I aimed to teach them something useful.

caring is meaningful even if there is no tangible reward

thanks mahatma

:)

:-) the sunnyvale starbucks folks were very friendly (and not hard on the eyes)

which is how i ended up with a card

hmm, maybe i should make a Gandhi proposal to them?

@Leslie: Apparently, you have been correct all along. "Geometry" really is garbage.

i used to live a short walk from a decent place. now i don't. they even closed the nearest starmuck during the pandemic

@TedShifrin this is what i've been trying to tell you.

my eyes are melting.

I really do not like Starmuck's coffee.

it's like raiders of the last ark over here.

i have a preference for dingy, dark, filled with coeds.

i drink chai :-)

Coeds ... that's a very dated (sexist?) term.

i am very dated

apologies

why are coeds girls

surely the co should mean either?

i did make that mistake early on, i went into a coed locker room thinking it was shared

i mean, i have been in places where even shower facilities are shared.

i presume it dates from when colleges first began admitting women.

i like younger company, regardless, more full of life :-)

my dad's (public!) college did not admit women when he went there.

wow.

guess what his (public!) high school didn't admit when he went there. :(

Entered college in 2010. Even then, we still had "co-ed" residence halls as opposed to male-only or female-only ones

mom's away camping, that means cheese steaks are on the menu.

Hi!

Is this a valid solution? Shouldn't the solution include cases $x\geq 3,0\leq x <3,-3\leq x<0,x<-3$?

how can $x<0$???

surely $|\text{anything whatsoever}| \ge 0$.

@copper.hat $x$ in $x^2-9$ not answer...

what does that mean?

the solution isn't how i would write it at all, but it does consider the case x \geq 3 when it analyzes what the equality means if x^2 - 9 \geq 0

similarly for the other cases

@copper.hat Why? $x\ne\mid \text{ anything }\mid$

the "if x^2 - 9 [blah]" lines are breaking down those cases

you must have $8x \ge 0$ so $x \ge 0$ off the bat.

you might also begin the analysis that way, although it's certainly not necessary to do so and maybe isn't simpler. the quadratic still leads you pretty naturally to further cases.

you are correct that it is $x \in \{1,9\}$ but you might as well go simple from the get go.

a little prose goes a long way with stuff like this. every time i see $\implies$ and $\or$ i cringe

whatever the or symbol is. i don't even want to know

@leslietownes But it doesn't include $x<0$

wolgwang, you can split the analysis into cases in a number of ways.

one way (based on the sign of x^2 - 9, which appears to be the approach taken above) does include analysis of negative numbers. it doesn't divide the cases at zero because that's not as relevant to the sign as x^2 - 9 as another condition might be.

but the conditions that are analyzed still cover all possibilities.

I would note that $x \ge 0$, then split into $[0,3], [3,\infty)$. The latter gives two quadratics and one of the solutions of each is negative so can be discarded.

@leslietownes Sign of $-8x$...

The case $0 < x < 3$ is handled when $x^2 - 9 < 0$. The case $-3 < x < 0$ is handled when $x^2 - 9 < 0$. All other cases are handled when $x^2 -9 \geq 0$.

a prose narrative of the approach above would begin "we consider two cases, depending upon whether x^2 - 9 is nonnegative or negative. The case that x^2 - 9 is nonnegative corresponds [some analysis] to values of x lying in [-3,3]. for these x, the condition simplifies to [blah, which we solve]. The case that x^2 - 9 is negative corresponds [some analysis] to values of x for which x < -3 or x > 3. For these x, the condition simplifies to [blah, which we solve]."

the sign of -8x doesn't need to explicitly enter the analysis because can be handled algebraically the same way, irrespective of what x is.

i swapped the conditions in my narrative but i hope the underlying point is clear.

i was dropped on my head as a child.

very early and very often

@leslietownes Yes

Thanks :-)

because i don't think ted is looking, if you are worried about getting lost in a problem like this, draw a picture. to graph |f(x)| graph f(x) first and then reflect the negative stuff over the x-axis. and then graph whatever you're comparing it to or setting it equal to. good sanity check. visually confirms what you'd see with algebra.

when i look at such things, i look for easy stuff first. to me the $x \ge 0$ is easy.

but i would strongly recommend what @leslietownes is suggesting, draw a picture.

we are visual people, as gandhi will tell you.

i think i'll revise my anti picture stance. i support drawing pictures in R^2 or R^1. but nowhere else.

give $\mathbb{R}^3$ a chance.

when they make 3d pens, sure.

i guess some of those VR headsets have paint programs that do give you a 3d pen.

i do use paper & glue fome time to time.

what about R^4 with a colormap to visualise a non-spatial dimension?

i'd have to huff quite a lot of glue before that made sense to me.

when i used to do optimisation we tried all sorts of visualisation tricks. colors, etc. but they require, in general, too much processing power to be useful in a commercial tool.

they work really well in some circumstances, obviously temperature, stress, etc.

I am so glad I didn't write my final master's project on statistical graphics

but for $\mathbb{R}^n$ interpretations of a design vector, useless.

besides, almost every problem i have worked on can initially be reduced to 2 and at most 3 dimensions

i don't mean final result, but early on.

I made a mind-bending animation of a 6-diminesional dataset once, where it was plotted in 2D at any given time, but then the animation kept "rotating" to new axes

it didn't help in the end

you could license that for use as a backdrop at raves

the problem is that it can help, but it needs to be case specific

I think it got lost to the aether a long time ago

it wasn't particularly sophisticated though,

generally a problem domain has already worked out the important quantities, at least for an initial approach.

even shape design (cars, flows, aeronautics) comes down to a few meaningful parameter (initially).

it may reflect the fact that humans are posing the problems in the first place :-)

copper hat designs cars with stone wheels that you operate with your feet. only a few meaningful parameters.

see an ex albany high school student of means was arrested for driving his tesla across the bay bridge from the back seat.

operated by his wallet

i saw that. i hope he doesn't injure anyone. it certainly seems like he's going to

@leslietownes How will I do that for $f(x)=\mid x^n-a \mid +bx$?

Sorry :-(

draw a parabolla

set reflect the part that should be negative

draw a line $bx$

imagine subtracting $bx$ from what you drew before

and try to draw the result with a bit aof a funky skew

@AndrewMicallef it is only a parabola if n=2.

note that something is the same. two natural cases are x^n - a >= 0 and x^n - a < 0. a difference is that it's no longer as easy, or perhaps possible, to solve x^n - a + bx = 0 or -(x^n - a) + bx = 0 by hand.

sorry true

also the precise way that something like x^n - a >= 0 boils down into a condition on x may further depend on the sign of a and god help us the parity of n.

@AndrewMicallef For example of $x^2-9 -8x$?

so maybe this lends itself more to counting solutions and not giving exact expressions for them.

I drew something like this...bare with while I fetch my camera

I really like this question, though I don't know how to solve it:

that hyphen is in the wrong place!!! make it stop!

actually the subtracting 8x is a bit tricky

Lol

interesting problem. my guess is it's ugly or has a ugly 'nice' answer involving analysis of 'special functions'

@Wolgwang if this equal to 0. Then you just need find the intersection btw $|x^n-a|$ and $-bx$

yeah my skecth looked nothing like this: wolframalpha.com/input/…

i was thinking more of sketching the absolute value thing and the non absolute value thing. wolframalpha.com/input/… maybe a little easier to see what's going on

@leslietownes Aghast! Pictures!

@Clarinetist you mentioned it in passing before, but what have you observed as the "jump" between doing your masters in stats vs pursuing your PhD right now?

Also salutations to the whole peanut gallery

:-)

if <v,Uv> is real for some unitary matrix U and some fixed vector v, is there a reasonable guess whether <v,UUv> is necessarily real, or could be complex ?

I've been experiencing that in my undergrad here....that's why I took a break from it....no longer treating things as the minimum

so in essence a math undergrad degree....lol

is it time constraints why you've chosen to stop? or value of it to you personally?

My undergrad - though admittedly a bit of a joke of a degree - very much stuck to its textbooks and prerequisites. I didn't come from a great program.

Time constraints primarily, yes. I mapped it out about a month ago and called it off when I noticed it would take at least 7-8 years

(part-time)

@A.M. Isn’t the first always real? For the second, is the square of a unitary matrix unitary?

yea that is a long commitment, because it will affect how much effort you can put into professional development out of that

<v,Uv> is not necessarily real. You may be thinking of <Uv,Uv>

No, I know what you said. Expand in terms of an orthonormal basis of eigenvectors.

Oh, sorry, I made one stupid mistake. The eigenvalues area unit complex numbers.

So the answer is generally no.

I was stupidly thinking hermitian but then didn’t follow through. Nevertheless, what I said is the best way to analyze the situation. Diagonalize $U$ with an orthonormal basis.

that's a good plan. For some reason, I was thinking there might be a slicker way to get what I need.

I believe it's easy to make a counterexample.

Start just with the diagonal case.

heya Ted

I got a question right up your ally

suppose the polar change of vars $(x_1, x_2) \mapsto (\cos \theta, \sin \theta)$. Then in particular, $|J| = 1$ (great)

what would be $\nabla f$ in the new coordinate system?

oh, we're working on the unit circle in the plane.

leave ted's allies alone.

:)

oh lordy.. Ted's got Brittany fans now

@JoeShmo Your Jacobian is nonsense.

You need to map one dimension to one dimension.

what do you mean?

And what do you think the “usual” gradient on the circle is?

the tangent line to the circle

What does that have to do with a function?

Anyhow, better to work in $\theta$ coordinates in the first place.

wait but I'm not following though

suppose $f : \mathbb{S}^1 \rightarrow \mathbb{R}$. Then $\nabla f = \begin{pmatrix} \partial_1 f \\ \partial_2 f \end{pmatrix}$. I should be able to come up with an expression in polar coordinates, no?

No. That formula assumes $f$ is defined on $\Bbb R^2$.

heh. why, because there is no limit on $\mathbb{S}^1$, say from the right?

Huh?

why does that formula assume that $f$ is defined on the entire plane?

second coord in gradient, i think

How does a function on the circle know how to extend to an open set so that partials make sense? This is the whole point of working on a manifold without working globally .

right, that's what I was getting at two comments above

limit = limit of difference quotients, for a fixed direction, say from the right

OK. I didn’t know what that meant.

I have a nasty tendency of assuming people can read my mind

The point is to work with the function and the metric in terms of $\theta$.

Quick question: Consider the function f(x,y)=2xy. What can we say about the convexity of the function f?

On what domain?

nothing is given

this is all I got

Well, what is your definition of convexity?

Hm... just the regular definition. I want to see if we can conclude anything for this? like it's neither convex nor concave?

I don't know what you think the regular definition is.

Do you understand why I asked about domain?

there are several conceivable equivalent definitions, but verifying that one is satisfied might be easier than verifying a randomly chosen one is satisfied

hands the ball to leslie

this is really copper hats thing. just like him to be conspicuously absent when convexity comes up.

$r=1$, always, meaning $f$ isn't a function of the radius, only of the angle (which is also why I was uncomfortable with the Jacobian. Technically the derivativie w.r.t $r$ should give me $0$)

i'm here :-)

@TedShifrin asked a question, I can do no better than repeat it...

@TedShifrin is it an open ball?

hello all, I am here to impress you with a question

uh

what's the definition of sin x when x is bigger than $\pi/2$

it's a closed ball, with no boundary, obviously.

But just from the outset, the function is $t \mapsto -2t^2$ on the subspace spanned by $(1,-1)$.

@shintuku $\sin(\pi-x)=\sin(x)$

ah, swell thank you

bunch of $\sin$ners

@shintuku $\sin(x+\pi)=-\sin(x)$

and that's not an identity, but a definition, right?

No, those are identities

then I've been lied to all my life when people told me sin was the ratio of the sides of a right triangle?

my convex engines are all revved up now

the definition depend on how the trig functions are developed

@shintuku no

sine is the y-coordinate of a point on a unit circle

if you rotate the hypotenuse about the center $20 \times 2 \pi$ times, did your triangle change?

@shintuku that is one development. The sign of the sine comes from the signed $y$ distance

bit of a tongue twister there...

for f(x,y)=2xy, can I say the principal minors of Hessian is non-negative so f is convex? Does this make sense?

joe, your original map does not seem to be a change of coordinate. the jacobian has a zero column

yes..

@JoeShmo well, the triangle wouldn't change but I would think if it was defined as an internal angle, $\sin 20 \times 2\pi$ would make no sense

@Quin see my last comment to Ted

@statwoman look at the function $\phi(t) = f(t,-t)$. Is that convex?

@shintuku it does make sense and it is $0$

right, I know from thinking about the unit circle

alright, well, the unit circle is good enough

thank you for the comments

yeah, i saw. if it restricts to the unit circle, then it is constant along level sets of $r=const$ but when someone says a function on $S^1$, it is usually implied that we are talking about a single-variable function that is periodic.

i'd also doublecheck that hessian.

it seems you are pulling back by inclusion of the circle into $R^2$

@statwoman The Hessian is trivial to compute and the eigenvalues are easy to compute.

bingo

yes, which was not my intention

the truth is that there is a stupid question at the heart of all this, with at least 2 known serious typos

maybe just look at f(1,-1), f(-1,1) and f(0,0). who needs t.

i think the convex question is coming from Elon's spaceship on the way to mars

i love tea

did somebody say dogecoin to the moon?

i always see "dodge-ee-coin", not something that engenders faith

let me just throw out a question and run.

you don't like your currency's value fluctuating with the mood of an eccentric tech CEO's twitter feed?

haven't quite adapted to contemporary interaction norms

that should be reason enough to suspect digital currencies that are not government backed.

there's a case to be made for a non government backed blockchain..

say, as an inflation check

(I realize the irony in the last statement)

that's where lesliecoin comes in.

it's a check on a lot of things. you won't regret buying it

as long as you bought it first

how do you cross things out? test

i was going to gamble $10k in 2014 on bitcoin but was put off by the 5% in & out transaction costs.

Madoff Lesliecoin

theranuscoin

hah..

good one^

d o g e c o i n

got reamed by liz?

not sure how he suddenly figured out the environmental impact. that has been well known for over a century now...

he suddenly figured out that he is the secret doge whale

he is a bit of a moby dick

and it would benefit him if he dramatically swapped out bitcoin for dogecoin as a payment method for his high tech cars, and he's got a sexy pretext for it, too

is suspect any block chain based mechanism has some environmental component

Are you using the usual (rotationally-invariant) metric on the circle, Shmo?

the economist this week is making a case for government digital currency

well a blockchain dollar I think is just a no brainer. why not? in fact, it already exists. but the fed's gotta control it's circulation

would that involve "mining"?

OK, Ted, here's the full claim -

ps i said some nonsense above and have an implication flipped, sorry

lmao

oh lawd

ooh dear

Matt Gaetz just felt the bern

digging in deep

dogeing deep

i censored myself

alright, let's be adults here

ok, i'll leave

no, stay. if you can behave

i am looking for a nice quote from douglas adams...

I'm here, typing out the problem statement for ted

elite english 19th C students referred to football as assoccer to distinguish it from rugby.

follow rugby?

im not sure if i should be surprised connacht beat munster

only when ireland is doing well

ahhh, that's different

munster it will be

Let $\mathcal{L} f(x) = -\dfrac{1}{2} x \cdot \nabla f(x) + \dfrac{1}{2} \nabla^2 f(x) - \dfrac{1}{2} x x^T : \nabla^2 f(x),\ \ x \in \mathbb{S}^1$.

The claim is that for the change of vars $\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix}$, $\mathcal{L}f = \dfrac{d^2}{d \theta ^2}$

which looks like it could make sense

its not convex, i'm leaving

:)

the first term disappears, $\cos \theta (-\sin \theta) + \cos \theta \sin \theta$ stuff

can't find my quote from The Salmon of Doubt

but, @TedShifrin, I guess $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ here, with $x \in \mathbb{S}^1$

That is not a change of variables … it's a parametrization of the circle. It's just sloppy. What's that colon at the end?

No, because for a general $f$ the first dot product needn't be zero.

I guess the last term is $$\dfrac{1}{2} x x^T : \nabla^2 f(x) = \dfrac{1}{2} \Sigma_{i, j \le 2}\ \ x_i x_j \partial_{i j} f(x)$$

What crap.

shouldn't that be a semicolon?

So what is $\nabla^2 f$ In the preceding term?

laplacian

yeah, I hate this dude's notation

I have no patience for this.

and theres A LOT of serious typos/mistakes in his book

He should be drawn and quartered.

Can I ask an elementary question related to Physics (mechanics) here?

@Silent You can ask anything. You may or may not get a worthwhile answer.

I'll take it you changed your mind, @Silent

elementary mechanics is euclidean geometry, so you'll get an answer

i had a call with my Google customer today, so i shaved, put on a shirt etc. however, everyone else on the call is unshaven/t-shirt/informal. am i really that out of date?

have basic formalities gone out the door?