Mathematics - 2022-10-07

Akiva Weinberger

12:00 AM

(specifically for a circle of radius 1)

OK eventually I'll run out of time to edit my comment

There it goes

'cause $|dz|=dz/(iz)$ for $z$ constrained to the unit circle

Ted Shifrin

1:02 AM

@Xander @robjohn I'm very annoyed that this OP deleted his post after our comments. My comment should probably have been an answer. I was only able to find it by going backwards on my own web browser. Can we do something about it?

robjohn

2:25 AM

@TedShifrin fixed

Add an answer if you wish

Ted Shifrin

@robjohn Many thanks.

Akiva Weinberger

3:10 AM

Fun fact, $\left\lvert\int_a^bf(x)\operatorname d\!x\right\rvert\le\int_a^b\left\lvert f(x)\right\rvert\operatorname d\!x$ isn't always true

leslie townes

i'll allow for the mathrm d, but that spacing is -- oh you fixed it

Akiva Weinberger

because, and this sounds extremely dumb, if $b<a$ then the integral on the right is negative but the one on the left is nonnegative

@leslietownes Yeah for some reason I sprung for the fancy TeX

Excuse me, $\TeX$

leslie townes

the thing with interpreting the intergral as oriented when b < a is a necessary evil in calculus class but in general i think of it as a bad thing

i've never been compatible with orientation

Akiva Weinberger

It's the tame version of the (actually serious) complex analysis issue I was talking about earlier

where students accidentally wrote statements equivalent to $|2\pi i|\le0$

leslie townes

it's like the change of variable formula when the map isn't one-to-one

monsters lurk within

Akiva Weinberger

3:13 AM

Ah yeah

This is why differential forms... something something, I think

That sentence got away from me

Here's a neat trick. Show that if $f(z)=a_nz^n+a_{n-1}z^{n-1}+\dotsb+a_1z+a_0$ then $\max_{|z|=1}f(x)\ge\max_ka_k$

My students (cowards) did this in one line with a special theorem from the textbook that essentially states exactly this

(which I'm pretty sure was the intended solution)

whereas actually proving from scratch is a good puzzle

(The problem actually was just to show $\max_{|z|=1}f(x)\ge1$ in the special case where $a_n=1$, but the more general thing also holds)

leslie townes

a_k are real?

Akiva Weinberger

Oh sorry $\max_{|z|=1}f(z)\ge\max_k|a_k|$

leslie townes

f(x) is real?

Ted Shifrin

applauds leslie obnoxiously

Akiva Weinberger

Look-

The version of what I said that makes sense is what I meant

Work with me

leslie townes

3:20 AM

i can do this all night

Ted Shifrin

But you're falling victim to all the stuff we agreed was very important for students to get right.

Akiva Weinberger

begrudgingly you have a point

I think I've been pretty lenient on stuff like that, to be honest. The only reason I took so much off from this person -

4 hours ago, by Akiva Weinberger

At least these weren't the person who tried showing that $\sup_{|z|=1}|e^{1/z}|$ was $e$ by writing, "By way of contradiction, suppose $|e^{1/z}|>e$. Then $\frac1z>1$. Then $z<1$. Then $|z|<|1|$. But this contradicts that we're on the unit circle."

- was because I couldn't find a quick way to repair the argument by assuming those were just typos

Ted Shifrin

I would not be lenient about thinking $\Bbb C$ is ordered.

Nor that $\log$ is single-valued. These are huge thematic points in $\Bbb C$ analysis.

Akiva Weinberger

Right. But if I could find a way to assume "oh they accidentally left off a magnitude bar" somewhere it might be different. But that argument is legitimately unrepairable

On this problem about the polynomial, a few people are concluding that $\sup_{|z|=1}f(z)\ge1$, and not adding the extra sentence that justifies $\sup=\max$

Ted Shifrin

I'm saying you shouldn't be so lenient even when you can repair.

If you don't nail them on homework, they'll mess up on exams and say "but the grader didn't care."

If the prof says to be lenient, then fine, but if I were the prof, I would order you not to be.

Akiva Weinberger

3:26 AM

Fine

I feel like not justifying $\sup=\max$ is a 0.5 point offense

which is enough for them to at least look at the problem and register that something happened

Ted Shifrin

My experience in 40 years of teaching (and grading most of my own homeworks beyond calculus-level) is that it's better to be tough on homework grading and slightly more lenient on exam-grading because exams go badly enough without that.

Grading out of 1 point, I agree that I wouldn't dock them half the problem for that.

I usually grade problems out of 3, 5, or 10, or something more divisible.

Xander Henderson

@TedShifrin I like 3.

Akiva Weinberger

I'm grading out of 10

Six problems, sixty points total for the p-set

Ted Shifrin

Well, I would dock more than .5 points out of 10, for sure.

Xander Henderson

3 = almost perfect; 2 = okay, but there are some important errors; 1 = well, at least you tried; 0 = blank or unintelligible.

Ted Shifrin

3:29 AM

Anyhow, I'm 8 years out of practice now.

Xander Henderson

Anywho, I just finished my evening class, and am going to go to bed now, because I have a conference to run in 10 hours.

G'night.

Akiva Weinberger

I'll discuss with the professor tomorrow before submitting the grades, and he can recalibrate me.

leslie townes

i never did fractional points. if it's worth marking off for, it's worth taking off real points.

there should also be a lot of room for scolding in red ink without taking off points.

Xander Henderson

Not red ink.

Green. Or purple.

Red ink is so angry. :(

Akiva Weinberger

I'm doing digital lol

Xander Henderson

3:31 AM

But I really am going to bed now.

Akiva Weinberger

Gradescope

Ted Shifrin

Night, Xander. Happy conference!

Akiva Weinberger

Ooh, some people are using the maximum modulus principle on $z^nf(1/z)$, which reverses the order of the coefficients

Ted Shifrin

@leslietownes only the best students read.

Akiva Weinberger

I really am learning more from my students than they are from me [cliche intensifies]

relevant: theonion.com/…

Ted Shifrin

3:34 AM

I used to spend countless hours grading and writing comments and suggestions. With many students I could tell they had read none.

leslie townes

i generally found that i couldn't expect someone to read something if i didn't take points off.

Akiva Weinberger

Yeah, that's my expectation

leslie townes

i usually managed it so you could get 9/10 on every homework and still get an A+ in the class.

Akiva Weinberger

though as a student I also have experienced receiving a failing or near-failing grade on something and thinking "oh god I can't bear to even look at this"

(Though that hasn't been relevant to my grading!)

(Also, not math classes.)

Ted Shifrin

Read the NYT about the organic chem prof who was fired at NYU because the woossy students complained.

Akiva Weinberger

3:36 AM

If you read the Reddit thread on that you'll get a very different perspective, lol

(Reddit has a lot of students on it)

Ted Shifrin

Oh hell, I had A students in upper courses with 80 averages on hmwk and exams.

Akiva Weinberger

If 80 averages count as As that's less of a problem I think

Or more of the opposite problem, which might have been your point

Ted Shifrin

I don’t believe in standard grading systems beyond low-level courses.

Akiva Weinberger

The students don't have a choice to not care about grades.

leslie townes

the worst thing i generally saw when TAing, not so much when teaching, was profs who would allow someone to get a C+ or even a B on an exam without correctly solving a single problem. partial credit run amok.

Ted Shifrin

3:39 AM

That doesn’t mean they deserve high ones just because a realistic grade keeps them out of grad school.

Akiva Weinberger

Fair enough.

Yeah, I may have to start being more strict with my grading. A friend of mine who's taking the class, and who doesn't know I'm grading him, mentioned to me the other day that he was surprised people were doing so well grade-wise

('cause they have access to min, med, max, mean, and st dev)

Ted Shifrin

This is not a new phenomenon. Back in the 70s a pushy Jewish mother called and harangued my dad at home !! Because the C her daughter had earned in Music Theory 3 might keep her out of grad school. My dad, stunned, responded, So?

Akiva Weinberger

and I joked back that I could "tell Ethan [the TA] to start grading more harshly" and he said "Oh no, forget I said anything"

leslie townes

i'm not super doctrinaire, but i do think, if you can't solve a single problem on an exam correctly, on midterm 1, midterm 2, and final, then you should not pass the class. and you're f*cking up the next prof's job if you, as an instructor, let them do it.

Akiva Weinberger

@TedShifrin They couldn't curve it to a B sharp?

@leslietownes That's fair. There's an old analogy - and I don't remember who came up with this - that teaching is like having someone build a foundation of a building, looking around, assessing, and stating "I think this is a C foundation... Now onto the first floor."

Ted Shifrin

3:42 AM

When I graded my intro to higher math course, leslie, my course grade was largely based on numbers of problems done right on the final. I.

think that for that sort of course you’re more than right.

leslie townes

i had a parent phone me once. they were a big wheel at the medical school (or wanted me to think that) i said, uh huh, uh huh, ok. call the chair if you feel that strongly about it. never heard of it again.

Akiva Weinberger

The music theory joke I made is messing with my head a bit because I never thought before about how in grades C is lower than B but in music it's higher

leslie townes

it's kind of... both, in music? isn't it?

Akiva Weinberger

If you go all the way around, I suppose

(B sharp and C are the same note, anyhow)

Ted Shifrin

@AkivaWeinberger We’ll just grade numerically, as in Europe.

@AkivaWeinberger Not with well-tempering.

Akiva Weinberger

3:45 AM

Wasn't there a thing where in some countries 1 was the best and 5 was the worst and in others it was the opposite

at least in decades past

Ted Shifrin

And MIT computes GPAs on a 5-pt scale, not 4.

Akiva Weinberger

@TedShifrin Right, I forgot...

JJ H.

Aops sure hire lots of people

Ted Shifrin

Never quite sure why, either as a student or as a faculty.

Akiva Weinberger

Octave equivalence is a sham anyway. Bohlen–Pierce all the way /s

Ted Shifrin

3:48 AM

@JJH. I’m not convinced they’re all of the highest caliber. I taught in person for them 2 years, before covid.

Akiva Weinberger

I was a counselor at BEAM over the summer, which is run (or at least funded? not sure) by AOPS

They're AOPS-affiliated somehow

Was really fun

(BEAM used to be called SPMPS back in the day, apparently)

(pronounced "spumps")

Little 11- and 12-year-olds learning about fractions

leslie townes

i've had a miniaturized version of this with some law schools having HH and variants as a qualified (i.e. not as good) version of H and others where it's better.

Akiva Weinberger

What's H?

Ted Shifrin

Indeed.

leslie townes

honors or high pass or something. maybe just H the same way KFC is KFC now.

Akiva Weinberger

3:51 AM

That's what outside of Germany they call a B, yeah?

(Another music joke)

H = ha! HH = ha! ha!

Ted Shifrin

(H majeur?)

leslie townes

so many law schools want to say on paper that they don't give letter grades when in reality they just use different symbols for A, B, C, D

Akiva Weinberger

Les françaises also have an H?

Tier lists online for whatever reason go S A B C D F

or maybe that's just TierZoo

Akiva Weinberger

4:09 AM

Ah

-10 points for "Did not solve problem"

Starts an induction proof, gets nowhere, draws a large X mark through the entire second page

leslie townes

large X mark through a page is very much a vibe i'm feeling

that should be worth positive points

Akiva Weinberger

I left a note on one line in the X'd-out page

saying that "I feel like you were kinda close here"

but they didn't do anything with it, so...

Like, "Oh, you wrote out the perfect theorem to use, then forgot about it"

leslie townes

vaughan jones once wrote "courage!" next to a point where i was doubting myself in an exam situation

Akiva Weinberger

Ha!

Of the Jones polynomial in knot theory?

Ted Shifrin

Maybe worth a few points, but the X ….

leslie townes

4:18 AM

yep

i couldn't quite remember the hypotheses of what i was trying to use and was frank about that in my exam paper, when i could probably have styled it out just by saying 'because X implies Y, then Z"

courage!

Ted Shifrin

Sounds like Julia Child’s advice …. When flipping a potato pancake, etc.

Akiva Weinberger

With enough confidence you can do anything.

leslie townes

maybe not quite that, but one does need confidence to do some things

Akiva Weinberger

except for my friend in my smooth manifolds class who asked me if writing "Obviously, $C^\infty(M_1\times\dotsb\times M_n)=C^\infty(M_1)\oplus\dotsb\oplus C^\infty(M_n)$" was enough justification or if he should replace the word 'obviously' with a few more sentences

leslie townes

and certainly one should not snitch on oneself in an exam paper by spelling out exactly what one is failing to remember

Akiva Weinberger

4:23 AM

whom Ted has already heard about

(The theorem is neither obvious nor true)

Ajay

Leslie you should make a song about "one"

Ever since I started chatting on here the word "one" or "oneself" kept popping up, in most of your messages

leslie townes

i did that on my qual too, i said straight up "this needs local convexity because of counterexample X but in the moment i don't see how i'm using that here"

cue attack wolves

ajay i do my best to universalize and depersonalize because my experience is the human experience

Akiva Weinberger

One should use 'one' when one wishes to use an impersonal pronoun. Two should use 'two' when two wishes to use an impeopleal pronoun

Wow, that joke didn't work in my head or in writing

I think the impersonal "you" can trip up non-native speakers 'cause they're not always taught about it

leslie townes

my daughter wrote out the numbers 1 to 100 today and while some of the 3s and 7s are backwards she wrote recognizable versions of all of the numbers

maybe i'll scan it tomorrow

U2 made a song about 'one'

copper.hat

4:38 AM

wow

not sure i could do that

saw u2 where they were just t1

Akiva Weinberger

fun fact: the www in website addresses is in fact sextuple-u

copper.hat

don't believe everything you read in reddit

Akiva Weinberger

I have an open mind

or an empty one

not sure

Of course...

copper.hat

i'm pink therefore i'm spam

Akiva Weinberger

I've never sum'd a day in my life

Cogito'd neither

copper.hat

4:51 AM

i canceled my scientific american decades ago

User1865345

New York University professor fired after students say his class was too hard.

These days firing a prof is easy. Complain the classes are hard. The grading is bad. That's it.

What a shame. The prof in question is highly experienced and author of many textbooks. Can't imagine such things. He is 84.

leslie townes

not to be ageist but i wonder how great the instruction is at 84. i think i would be terrible at that point.

Akiva Weinberger

I think there isn't enough in the article to judge, to be honest

It really depends on how the class was

Also, it's notable that the student petition did not request that the prof be fired

User1865345

Yeh. NYU took that decision.

But the whole saga could have been handled better, imo.

Akiva Weinberger

I could imagine the class being hard because the teacher wasn't teaching, and I could imagine the class being hard because the students weren't putting in the required effort

User1865345

5:06 AM

@leslietownes this could be legit but again I have seen teachers at that age not losing the agility yet.

leslie townes

i'd 1000% agree with students at a place like NYU just not wanting to be bothered with instruction, but i'm hesitant to go immediately there because those are exactly my priors.

i suspect myself.

Akiva Weinberger

Also I've never taken orgo so I don't really know how hard the class is generally

I hear orgo is hard everywhere

My main thing is math, but nearly all the math I know is self-taught, so I'm not a product of the education system

leslie townes

it is one of the main weeder classes for medical school. non-organic chemistry usually precedes it and there are one or more math classes in the mix too.

Akiva Weinberger

People criticize how calc 1 classes are taught and my only response is "I have no idea" lol

(It's the time of night where I am an unrestrained braggart. Good to know)

(Maybe I should stop grading)

User1865345

@AkivaWeinberger that is great. Like George Green.

Akiva Weinberger

5:10 AM

Well ok I did go to school

so not that level of self-taught

User1865345

Lol. That dude was a legend.

There are many self-taught in the history. But this man took it to a new level.

Akiva Weinberger

Easier with the Internet now.

Do you know about the SoME2 contest that just ended? It generated a treasure trove of educational math content

Have you seen more math videos in your feed recently? (SoME2 results)

►

(By "self-taught" I mean "My dad bought me a ton of books")

("and I was the nerdy middle-grader walking around with a math book everywhere")

User1865345

5:48 AM

@AkivaWeinberger Good books serve the best. That's for sure.

2

Koro

6:16 AM

If $f_n$ is a sequence of $\mathbb {\bar R}$ valued measurable functions, then the limit $\lim_n f_n$ if exists is measurable?

If the the measure is complete, then yes.

But otherwise?

copper.hat

6:39 AM

If the limit exists everywhere then yes.

The limsup, liminf are measurable.

User1865345

7:28 AM

31

Q: Published AI-generated nonsense math papers

I guess most of us know that one can easily automatically generate a math-like nonsense paper, and that it is possible to have such a paper published. However, I was quite sure that nobody actually does that. But recently I was looking at papers which cite one of my colleague's articles, and I en...

Koro

8:29 AM

@copper.hat yes, I understood. Thanks a lot :).

Koro

12:40 PM

0

Q: Proving in details proposition 2.12 of Folland

Let $(X, M,m)$ be a measure space and let $(X, \overline M ,\overline m)$ be its completion. If $f$ is an $\overline M$-measurable function on $X$, there is an $M$-measurable function $g$ such that $f=g\,\, \, \overline m$-almost everywhere. Some symbols have been changed for brevity. To prove t...

real-analysis measure-theory solution-verification

ILikeMathematics

When you want to search up a question on math.stackexchange, do you usually have better results with Google or ApproachZero?

Koro

imho, searchonmath used to give the best results but it has partly become a paid service now.

Between Google and Approach0, I'd say the latter is better for search on mathstackexchange as the former doesn't seem (I'm not sure) to understand mathjax.

ILikeMathematics

1:57 PM

@Koro Google seems to give better results when you are searching for something that doesn't contain math expressions
As an example, take interesting Pythagorean Theorem proofs, searching "interesting Pythagorean Theorem proofs site:math.stackexchange.com" seems to give more results that are relevant than Approach0 (atleast on the first page)

Are you typing the keywords into Approach0 seperately? That's how I'm doing it, not sure if there is a better way to search only titles or something

user1027216

4:22 PM

Hi, I question. Is there a problem with that line? \begin{align*}
\log(n+1)-\left(\log(n)-1+\mathcal{O}(\log n)\right)=&\log\left( \frac{1+1/n}{1}\right)-\left(-1+\mathcal{O}(\log n)\right),\\
&\to 1,\quad n\to+\infty.
\end{align*}

Since that $\mathcal{O}(\log n)\to 0$ as $n\to +\infty$, so the limit should be $1$.

Koro

@ILikeMathematics I type equations and some keywords too.

Can anyone please take a look at the proof here?

0

Q: Proving in detail proposition 2.12 of Folland

Let $(X, M,m)$ be a measure space and let $(X, \overline M ,\overline m)$ be its completion. If $f$ is an $\overline M$-measurable function on $X$, there is an $M$-measurable function $g$ such that $f=g\,\, \, \overline m$-almost everywhere. Some symbols have been changed for brevity. To prove t...

real-analysis measure-theory solution-verification

Oliver Díaz

@Koro Just did. I left some comments in your posting.

@leslietownes U@ also made a song related to 2 in their seminal album War (Two hearts ...)

Koro

@OliverDíaz thanks a lot :-). After your comment, I realized that I should have said $c_j\ne 0$ in my post. I have made this small change in the post.

ILikeMathematics

4:37 PM

@Koro How did you link your question like that?

Koro

@ILikeMathematics Just copy url of the question and paste it here.

ILikeMathematics

Alr, thx

Could anyone please take a look at this, I don't seem to be getting any answers, I'm not sure if there is anything I should edit into the question
https://math.stackexchange.com/questions/4538686/how-to-determine-at-which-point-a-function-with-even-and-odd-exponents-is-symmet

I think my topic tags are pretty broad, yet it seems like not many people are seeing my question?

Ted Shifrin

5:01 PM

The analysis/real-analysis tags don't seem appropriate.

I would recommend doing a general translation substitution $x\rightsquigarrow x-a$.

leslie townes

use the one tag that is always appropriate: complex-geometry

Ted Shifrin

5:24 PM

slaps leslie

si-LV-er_and_b-LA-ck

:: offers other cheek::

🙏

☮️

user1027216

6:01 PM

A question related by my another question is if the notation $f(n)O(\log n)$ make sense?

ILikeMathematics

6:15 PM

Oh, alr, I will remove them then
But is the question hard/is there no good general answer? It has 82 views

Yai0Phah

6:35 PM

@AkivaWeinberger I forget the correct answer. They should be injective tensor products or projective?

Jakobian

I've finally found out that I'm doing math better when I'm not sleep deprived

Friday. What a blessed day

Taking your mind off of those stressful moments is surely a good thing

Ted Shifrin

6:51 PM

@ILikeMathematics Did you work out my suggestion about letting $u=x-a$ and considering $g(u)=f(a+u)$?

ILikeMathematics

7:09 PM

@TedShifrin Oh, I thought that was meant for Koro, let me try

ILikeMathematics

7:22 PM

@TedShifrin Can you please elaborate what $a$ and $f(a + u)$ should represent? I have $f(x) = 2x^3 - 6x^2 + 8$ and I want to check it for point symmetry at some point

Koro

Given $f,g:X\to \overline {\mathbb R}$ measurable, how to show that fg is measurable?

If it were $\mathbb R$ instead of R bar, then I can do this.

Ted Shifrin

So I'm suggesting you try to find the right $a$ if there is one. For example, is $g$ is even? Is $g$ odd?

Koro

Also, I think that defining a function to have value $\infty$ is nonsense.

leslie townes

koro i must be missing something, it's the composition of x -> (f(x), g(x)) with the continuous [hence measurable] product? who cares about infty?

Koro

Is there any rational explanation to this?

Ted Shifrin

7:26 PM

@ILike: For example, to say $g$ is odd (even is clearly hopeless) is to say $$4+a^3+3au^2=3(a^2+u^2) \quad\text{for all }u.$$

Jakobian

At least we're not using the one-point compactification of rational numbers

Ted Shifrin

@ILike Most likely you're going to need a vertical shift as well.

Koro

Leslie: I define extended Borel sigma algebra on $\overline {\mathbb R}$ as $\{B\cup H: B\in \mathfrak B, H\subset \{\pm \infty\}\}$. So I am trying to show that $(fg)^{-1}(B\cup H)$ is in sigma algebra on X.

In case of no bar on R, I showed $(f\pm g)^2, c f$ for any constant c to be measurable so $\frac 14\{(f+g)^2-(f-g)^2\}=fg$ is measurable.

Ted Shifrin

If I set $g(u)=f(a+u)+b$, what must $a$ and $b$ be in order to have $g(-u)=-g(u)$?

ILikeMathematics

@TedShifrin So you basically would try to factor $f(x)$ so you can see the horizontal and vertical shift?

Ted Shifrin

7:32 PM

No, certainly not factoring. Simplify the algebra when you do the shift.

Semiclassical

so here's a borinng linear algebra question i'm tryingn to remember how best to resolve

Ted Shifrin

Maybe you want to factor $g(-u)+g(u)$.

Well, hello, @Semiclassic, stranger.

o/

\o

\o/

7:33 PM

\o/

Ted Shifrin

Is that a picture of Trompolini on his golden throne?

Akiva Weinberger

incredible.pm

This is super tedious but kinda addicting

Ted Shifrin

So, DogAteMy, did the prof want you to be so lenient?

Semiclassical

let $u,v$ be a pair of oblique vectors in the plane (so not parallel but not orthogonal either) from which I form the matrix $M=uu^\top+vv^\top$.

Akiva Weinberger

Will talk in half an hour

Ted Shifrin

7:34 PM

Ah.

ILikeMathematics

@TedShifrin don't $a$ and $b$ have to be 0? $g(-u) = -g(u)$ means it's point symmetrical to (0|0)

Semiclassical

now, using the definition of rank I know that this is at most rank 2. is it possible that $M$ is actually rank one?

Ted Shifrin

Only if your original $f(x)$ was symmetric about $0$.

Semiclassical

i think the answer is no but i'm struggling to remember how to actually prove it

Ted Shifrin

@Semiclassic Only if $u,v$ are linearly dependent, I believe.

Semiclassical

7:36 PM

yeah, which "not parallel" would exclude

the grungy proof i guess is to choose wlog $u=e_1$ and $v=v_1 e_1+v_2 e_2$

Ted Shifrin

Let's assume $u=(1,0)$ and $v=(\cos\theta,\sin\theta)$.

But I want to assume they're unit vectors first. Then how can you have a kernel? You need $\text{proj}_ux = -\text{proj}_v x$.

But this immediately says $u,v$ linearly dependent.

Unless $x=0$.

QED

Semiclassical

hmm. what if they're not unit vectors? (or, equivalently, leave them as unit vectors but take $M=uu^\top + c vv^\top$)

Ted Shifrin

Same proof.

Apply to $x$. The first term is a scalar multiple of $u$, the second a scalar multiple of $v$.

Semiclassical

ah yeah

Ted Shifrin

Looking at the image is harder. Kernel is easier.

Semiclassical

7:42 PM

yeah, $(u\cdot x)u = -c (v\cdot x)v$ is convincing

Ted Shifrin

Certainly $x$ can't be orthogonal to both $u$ and $v$ :P Unless it's $0$.

Semiclassical

more generally that would mean that a sum of $k$ projectors is rank $k$ unless there's a linear dependence between the vectors

Ted Shifrin

among?

Semiclassical

among, yes

my students had a quantum computing quiz this week, where the question asked them to take some 2-by-2 real symmetric matrices, put them in the form $\sum_{k} p_k v_k v_k^\top$, and deduce whether they were rank one

Ted Shifrin

Ah, understanding the spectral theorem physically.

Semiclassical

7:46 PM

the intent was for the $v_k$'s to all be orthogonal, in which case the rank is simple to see

Ted Shifrin

Well, if you apply the spectral theorem, they are orthogonal.

Semiclassical

yeah, you can always find an orthogonal basis

but we weren't clear about that in the wording of the problem, so not everyone did

and the question i had was whether you could still use that to distinguish rank one or not

Ted Shifrin

Ah, you can do completing the square instead and get the $LDL^\top$ decomposition.

See chapter 5 section 3 of my book (or something).

Semiclassical

yeah

Ted Shifrin

This comes back to Sylvester's law, which we've discussed in here numerous times.

Jakobian

7:48 PM

@Koro what prevents you from writing $(fg)^{-1}(\infty) = f^{-1}(0, \infty]\cap g^{-1}(\infty) \cup f^{-1}[-\infty, 0)\cap g^{-1}(-\infty) \cup ...$?

Semiclassical

nice

Ted Shifrin

It all depends whether you're thinking of the symmetric matrix as a bilinear form or as a linear map. :)

Semiclassical

we don't actually use the word 'rank' in the course, to be precise. for us it's just a question of "can you write it as $M=a vv^\top$ or not"

if it is it's a pure state, if not it's mixed

Ted Shifrin

We can't assume quantum students know elementary linear algebra? SAD.

Semiclassical

lol

si-LV-er_and_b-LA-ck

7:49 PM

@Semiclassical how were the grades on the quiz?

Semiclassical

hasn't been fully graded yet

Ted Shifrin

Anyhow, I'm fine with the completing the square approach.

It ties in nicely with row ops and echelon form :)

leslie townes

ted i reviewed an article for a journal that had 'quantum' in the title and i said, i think you will find that if translated out of quantum language a guy named schur did this in 1920.

Semiclassical

nice

Ted Shifrin

@leslie Jargon hides things.

leslie townes

7:50 PM

it was a cool result though.

Ted Shifrin

OK, getting to be lunchtime for this bonzo.

si-LV-er_and_b-LA-ck

cya

Ted Shifrin

@Semiclassic Have I sufficiently addressed your concerns? :D

@leslie I don't know if you have something further to comment here.

Semiclassical

yeah

Goku

8:08 PM

Doing math during lunch break from coding

Koro

@Jakobian yup, that works. Thanks :-).

Let $B\cup H\in \mathfrak B_{\overline{\mathbb R}}$ and consider $(fg)^{-1}(B\cup H)=(fg)^{-1}(B)\cup (fg)^{-1}(H)$.

If $H=\{\infty\}$, then $(fg)^{-1}(H)=(f^{-1}(-\infty, 0)\cap g^{-1}(-\infty))\cup (g^{-1}(-\infty, 0)\cap f^{-1}(-\infty))\cup (f^{-1}(0,\infty)\cap g^{-1}(\infty))\cup (g^{-1}(0,\infty)\cap f^{-1}(\infty))\cup (f^{-1}(\infty)\cap g^{-1}(\infty))\cup (f^{-1}(-\infty)\cap g^{-1}(-\infty)) $

Similarly, the cases when $H=\{-\infty\}$ or $H=\{-\infty, \infty\}$ can be handled.

This completes the proof.

si-LV-er_and_b-LA-ck

38 mins ago, by Ted Shifrin

QED

ILikeMathematics

9:16 PM

@TedShifrin But what's the point of considering $g(u) = f(a + u)$? Finding the horizontal shift?

Ted Shifrin

Yes, to find the $a$ which (possibly) gives you symmetry.

geocalc33

9:32 PM

This is my favorite matrix: $M_s=\begin{bmatrix}e^s&0&0\\0&e^{-s}&0\\0&0&e^{-s}\end{bmatrix}$

Semiclassical

ok. y?

geocalc33

because (correct me if I'm wrong) it's a generalisation of $A_s=\pmatrix{e^s&0\\ 0&e^{-s}}$

I also like the simplicity of $M_s$

Akiva Weinberger

10:16 PM

Students did poorly :(

Out of 60: Min 24, med 55, max 60, mean 51.18, stdev 8.18

The person who got 24 arguably shoulda got less but I felt pity

@TedShifrin

That one person has been on our radar for a while

I think that student got the lowest score for all four p-sets so far

si-LV-er_and_b-LA-ck

How many 60s?

Akiva Weinberger

Uh

Two

Ted Shifrin

Median 55 is ridiculously high.

si-LV-er_and_b-LA-ck

^

Ted Shifrin

I don't consider this doing poorly, except for the poor 24. I agree that person is in trouble.

si-LV-er_and_b-LA-ck

10:23 PM

What was the mode?

Ted Shifrin

Just out of curiosity, I'm going to look up my homework grades from the last time I taught the graduate complex analysis course.

The average grade was under 69%.

The homework average in the last Honors Multivariable Math I taught was 79% (proofs only — computations on WebWork). That was graded by a former student of mine, not by me. The non-honors students averaged 52% on homework :(

In the second semester, the homework average was about 80%. 8/12 students got As in the course.

That was probably the all-round strongest class I ever had for this course; no super-super-stars but very strong.

Anyhow, your 85% homework mean seems high to me.

Great news: Mehmet Oz spoke at a very ritzy Rethugnican fund-raiser in Southern CA last night and stood in front of a car used by — guess who — Hitler, complete with swastika on it!!! Unbelievable!

Semiclassical

par for the coourse, really

Ted Shifrin

The homework scores or the last thing I just posted?

Source.

Semiclassical

10:42 PM

@TedShifrin last thing

Ted Shifrin

I still get gobsmacked by these pieces of garbage.

si-LV-er_and_b-LA-ck

It's all about the thug life.

> Thug life is a term used with pride, to describe a person who started out with nothing and built themselves up to be something.

The problem is they forget where they came from.

Ted Shifrin

That must be an urban dictionary definition which is different from the meaning I'm used to.

si-LV-er_and_b-LA-ck

They said it was misinterpreted as the gangster life.

Ted Shifrin

Maybe there are different definitions for people over 50.

si-LV-er_and_b-LA-ck

10:58 PM

dictionary.com/e/slang/thug-life

Ted Shifrin

In my old-person brain, there were no racial connotations to "thug." I pictured rednecks.

si-LV-er_and_b-LA-ck

Yeah, Al Capone thugs.

> 1810, "member of a gang of murderers and robbers in India who strangled their victims," from Marathi thag, thak "cheat, swindler," Hindi thag, perhaps from Sanskrit sthaga-s "cunning, fraudulent,"...

Transcript for

$Mathematics$ Mathematics