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12:31 AM
@Derivative Is $n\\Bbb Z$ or $\Bbb Z[i]$? Maybe think quotient ring first?
2 hours later…
2:08 AM
@geocalc33 how is $\Bbb R_{>0}^2$ multiplicative?
1 hour later…
3:30 AM
If f is continuous on [0,1], then how do I show that the sequence $I_n:= (\int_0^1 |f|^n)^\frac 1n$ is Cauchy?
it is not allowed to use the fact that the sequence $I_n$ converges to max f.
Hölder has to show up.
iirc you can show its bounded and increasing hence convergent hence cauchy @koro
essentially via Holder yes
3:46 AM
@CalvinKhor bounded is Okay: by continuity |f| attains its maximum M so $I_n\le M$
well. ok sure
Are you sure about monotonicity though?
attaining it is not so important, its that it has a maximum
@CalvinKhor true.
well i was informed in this chat a while back :P but I believe so
ill give it a quick check since i know how it should go...
3:48 AM
Or, let's just consider $g:[0,1]\to R^+$ instead of $f$ as we have $|f|$ at the right places.
yes, you can remove the mod signs. Also I_n ≥0
@CalvinKhor about what? The fact that attainment of maximum value not required?
no, that it was monotonic
i was "essentially" told this. as in. it was implied by a related statement
@CalvinKhor I see. :)
Anyways, I'll try proving it monotonic then.
and I don't see how Holder will help here. I think you meant Cauchy Schwarz (a special case of Holder's).
no, specifically holder
with p dependent on n. Or Jensen's
3:53 AM
I was wrong. No Cauchy Schwarz
ok, yes it is monotone
4:21 AM
How fare the nerds?
@Fargle Sun is blazing hot atm. Gonna have a glasses tan
Fun stuff. Nighttime here, but it's been ridiculously hot and humid where I am. Outside is almost a sauna.
4:37 AM
As Maude used to say, “God will get you for that, Walter.” God has had it with the US and the world.
Interesting living in a part of the US that may well become borderline, if not outright, uninhabitable over my lifetime.
5:37 AM
Anyone got linear algebra by Meckes and meckes?
There is a mistake in page 4.
176 cups is incompatible with teaspoon.
so it should be 176×48 teaspoon.
Please correct me if I am wrong.
@Fargle While the coasts will be submerged, I suspect. BTW, hi, Fargle.
we're gonna break off and sink into the ocean.
we almost bought a house right on the beach. we had an offer accepted subject to conditions that ended up not being met. it probably had 10 more years of being on the beach.
Prepositions are important. You don't want to buy property in the beach.
we didn't want to buy a starter kit for a coral reef and scuba wildlife sanctuary.
well, i guess we did, in that we did put in an offer. it would have been a good, beachy 10 years.
we live on a hill high enough that i think it will not be submerged in my lifetime, although i haven't researched whether it might become an island
5:55 AM
We've seen just a few days ago the news stories of a newly purchased beach home washed away on the NC shore.
oh yeah, the outer banks. that stuff is sea level already.
@Fargle: I forget if I told you or not. But Ken is now a daddy.
my late uncle used to vacation there and the place he vacationed is, i think, no longer reliably above sea level all year.
I have been lots of places on the east coast, but never to the NC seashore, let alone the outer banks.
it's alarmingly flat and only inches above sea level.
it would surprise me if you could get homeowners insurance for a place there these days
6:00 AM
@CalvinKhor Actually today would be the first time I'd learn Jensen's inequality for integrals.
markets already know what the politicians don't. try to get a loan to develop a ski lodge in a currently snowy area that is not really up in the clouds and find out.
I never noticed this inequality before. Thanks a lot.
koro: hadn't we talked about this? wasn't it on your sheet?
@Koro I take it you solved it, glad to help!
almost all of that contest crap on that formula sheet generalizes to integrals. it's the integral stuff applied to counting measure.
this is a weakness of wikipedia, i think. the standard format for a page on an inequality is five or six cases of something discrete, with applications and maybe proofs, and then something at the end maybe about integrals.
6:03 AM
Leslie: for integrals? I think it wasn't there in this form: if $\phi$ is convex then $\phi (\int f)\le \int \phi(f)$.
there's some normalizing there. that's f on a space of measure 1, i think.
it probably wasn't on the formula sheet, but, i would have thought we'd talked about it. maybe not. i could be blending this memory with something else.
If it is a probability measure
the discrete stuff can be used to prove the integral stuff, so it's not like you need special technology to get from one to the other.
but, hrm. i wonder who i was talking to.
@leslietownes I understand that. Holder for discrete for e.g. transforms nicely to integral form too.
@Koro if you know holder inequality in the form $\|fg\|_p \le \|f\|_q \|g\|_r$ holds when $\frac1p = \frac1q + \frac1r$ then the proof is one line
just from $\frac1n=\frac1{n+1} + \frac1{n(n+1)}$, $\|f\cdot 1\|_n \le \| f \|_{n+1} \|1\|_{n(n+1)} = \|f\|_{n+1}$
6:22 AM
figures that mr. convex would come in for jensen's inequality
chinese word for convex is literally 凸
an extended middle finger to those who would have thought it would have been convex
concave is 凹
that's awesome
Let $K$ be a closed set, and let $K^凸$ be its convex hull....
I suppose 凸 could also be copper.hat's hat
6:29 AM
it looks a bit like a devo energy dome
6:49 AM
clicking links in internet explorer is opening them in edge browser.
you have both IE and edge installed?
@CalvinKhor I wasn't familiar with that until now. I know the inequality when $1/p+1/q=1$
@CalvinKhor in my office laptop, they have kept both.
some intranet sites here don't open except in ie.
@Koro Step one: for any $a,b<\infty$, $f\ge0$, $ \|f^a\|_{b} = \|f\|_{ab}^a$
step two: if $\frac1p = \frac1q + \frac1r$ then $1 = \frac1{q/p} + \frac1{r/p}$.
oh i remembered it slightly off. but anyway
yeah, I'll try that. Thanks :)
oh no its fine
6:53 AM
no microsoft, i don't need to install windows 11, this year is already going badly enough
Step 3: $\|fg\|^p_p = \|f^pg^p\|_1 \le \|f^p \|_{q/p} \|g^p\|_{r/p} = \|f\|^p_q \|g\|^p_r $
dangit no preview
phew fixed it under 2m
I think I've got the twin prime conjecture now
Annals, how inappropriate for this site :P
why inappropriate lol
7:04 AM
Why not call it Rosebuds of Mathematics
Sounds much nicer than what they have put
Oh, shit, I'm a bad speller lol
I would definitely just arxiv my crap, instead of going annal with it
its not one or the other
I'm joking, lol
same m8
7:09 AM
Do you like twin primes?
not as much as you for sure, i work in analysis
Now that one is spelled with one n
Hey Calvin I need your feedback.
I want the question on MSE to be reopened.
Sure link us
we'll reopen that chit, pending we agree
and we're not being too anal-retentive
7:12 AM
The problem is that their justification for closing the question is not applicable to my case.
What was the reason?
Here is the question:
Q: What are some nice topics in undergraduate mathematics to write about?

bumblebeeAs a high school student, I will soon be going on to do the IB(college equivalent). As part of the IB curriculum we are expected to write an Extended Essay(EE) that counts towards our grade. For the EE we are expected to motive a question on a subject, or particular topic, and write a paper answe...

Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances.
Yes, I agree with you
I voted
i'm not sure that the quoted reason (if that's what it is) fully makes sense, but questions that are "too" open-ended do get closed with regularity.
hm. this is definitely on the border
7:14 AM
questions that fundamentally ask "hey, what do you think" get closed, and i understand why.
I think the site should allow us to ask the difficult questions
But i'm not asking for an opinion.
surely topic ideas are not opinions
"nice topics" is definitely asking for an opinion. i understand the purpose of the question.
Oh yes, don't think, don't get creative, just follow instructions from the textbook, thx
7:15 AM
it's a great question, but i also see why it got closed.
for the record, i did not vote to close.
Any recommendations for what I should to get it reopened?
Removing the nice part is definitely one.
maybe narrow it down to a topic that you have specific interest in. remove inherently qualitative adjectives like "reasonable" and "interesting".
if someone were to write an undergrad-level intro to topic X, what resources would assist them to do it, i am aware of X,Y,Z but want more on W.
Can we talk about Prime Twins for a minute?
i have no control over the harpies who vote to close, but, focus is always better. very cabined value judgments are less likely to attract the harpies than something open-ended.
I think all posts of this site should be deleted, they're all pretty bad
7:20 AM
to be fair, closure isn't deletion. closure means, this is in the realm of the site, but let there be no more answers to this until there is something fixed with the question. deletion is more for stuff that just doesn't belong.
I think you should get some leeway
did the comment about being confused if its highschool or ug level come before or after you explained that you are highschool but looking for higher level stuff?
maybe a "off-topic post allowance" that recharges when you ask or answer some good questions
the main response to any closed question is: focus, focus, focus.
It came before Calvin.
7:22 AM
@TedShifrin I believe you did mention that! I'm quite happy for him.
THat's what she said
OK then I would say you have already addressed that point
But I made sure to make it more clear by responding directly and editing the post/
@leslietownes closure is a prerequisite for deletion though, right? (if score>0 or something i cant recall)
ask a moderator. i don't know why i've gone out on a limb here opining about closure and deletion. i have no idea what mods do.
i'm just yelling into the chasm, same as the rest of us.
7:25 AM
I changed the first point to "Are there some resources that would be useful in explaining the topic like youtube videos, papers, journals et cetera?"
I removed the "as a note to the first point" part.
I think that should be quite obvious.
The moderators are just telling you to go study math and come back and ask about solutions to an equation, don't get social with us
@DiagramChasingBlues do you need more pot?
Don't get smaht
Then what are you doing?
I voted to reopen because I don't think its as broad as it ostensibly seems. Sure you could write an essay on e.g. wiles' proof of fermats last theorem but no one in the intended audience is going to learn anything. There are only so many hallmark results that are in reaching grasp of a highschool student. And if you know of a obscure result that is also interesting but not too technical, I would love to read about it.
7:31 AM
Okay, let me run this by you guys. Define $M = \{(x,y,z) : x - y - z = 0; x,y,z\in \Bbb{Z}\}$ clearly a $\Bbb{Z}$-submod of $\Bbb{Z}^3$. Follow me so far?
what does submod even mean?
yes i guess so
lol 'duh' he's a highschool student
7:34 AM
Don't worry, in 2 years i'll be in uni.
Okay, anyway. Can't we endow $M$ with a ring structure?
$M = \{(x,y,x-y) : x,y \in \Bbb{Z}\}$.
is another way of writing it
Define $(x,y,x-y)\cdot(a,b, a-b) = (xa + yb, xb + ya, xa + yb - (xb + ya))$
Do we need well-definedness here?
*I mean do we need to show it
I don't think we do
Anyway, is it clear to you that $\cdot$ is both associative and distributive over $+$ of $M$?
Well, how could it fail to be well-defined? You don't have anything here that's some sort of a representative of a class, so...
no, i would need some paper
Okay, cool, I will prove it in my post. They done donvoted my first sketch write
i dont like the minus sign, i'm guessing distributive might fail
7:39 AM
It's all linear though...
Okay, perhaps that's where the flaw was
I will check it on paper
SymPy would actually work for this too
well i had to take 5 seconds to remember what distributive even meant, so don't trust me
Distributivity seems to work. I suppose what I'd question here is: what, really, is the point of having it be $\{(x, y, x - y)\} \subset \Bbb Z^3$? Seems like this would wind up just being ring-isomorphic to $\{(x, y)\} \subset \Bbb Z^2$ with the appropriate multiplication.
yeah ok i was being silly. i think it distributes
Oh btw @CalvinKhor the final version of my EE will be read by a math professor somewhere in the world. So you are right, the intended audience will not learn anything from my EE.
Is that the grader?
7:47 AM
Unless I write about some super shady thing no-one knows about.
Yeah, they are the grader.
the grader is not supposed to be the audience, they are supposed to grade according to the rules
Indeed, this just looks like the split-complex integers $\{m + nj\;|\;m,n \in \Bbb Z;\;j^2 = 1\}$.
@CalvinKhor then are criteria, rules?
if you wanted to write to teach a math prof something maybe tell him how to do his taxes or something not math
But I don't know how much the tax bracket is.
I don't even know them
They won't even know me
7:50 AM
in Ajay personal room, 6 hours ago, by bumblebee
It must be written so that fellow students can understand
@DiagramChasingBlues So, in short, the answer is yes, and in fact it's an already-known ring structure, just with some extra baggage in that third coordinate.
I read this to mean that your fellow students are the target audience
The "extra baggage" can be simplified using the first form of $M$
They are test subjects to see if your work has been communicated well
i.e. Math HL students.
it's then just $czw$ instead of the sum thing
7:51 AM
I also forgot to mention that math teachers will be checking.
well i disagree, I do not want high school students to help me check what to read. But I only have my opinion. I would say check with a teacher
I agree.
and my opinion should not be very important, because I dont know anything about IB or the IB EE
@DiagramChasingBlues Well, what I mean is, because of how you've defined the submodule, you don't actually need that third coordinate, which can be seen because it's isomorphic to the split-complex integers (as a ring structure on $\Bbb Z^2$), and indeed simply by the projection map $(x, y, z) \mapsto (x, y)$.
Though, I do have a friend who has some insane math roots. Her grandfather(When alive) and father write research papers heavily concentrated with math.
Michael Lvovitch Tsetlin (the surname is also written Cetlin, Tzetlin, Zeitlin, Zetlin; cyrillic: Михаил Львович Цетлин) (22 September 1924 – 30 May 1966) was a Soviet mathematician and physicist who worked on cybernetics. He introduced the Gelfand–Tsetlin basis for finite-dimensional representations of classical groups. He was the founder of the learning automaton approach to machine learning. == See also == Gelfand–Tsetlin integrable system Boolean differential calculus Learning automaton Tsetlin machine == Collection of the publications == Tsetlin M.L. Automation theory and modeling ...
I wonder if I can get her dad to check 😏
7:56 AM
What I mean is, check with a teacher who the "intended audience" should be
Like, if my teacher says, write an essay for 5 year olds to read, a 5 year old is not going to be the person marking the work
....sorry for typo
@bumblebee co-author with gelfand is the hugest casual name drop ive seen in a while
i don't know who Gelfand is.
weird flex but OK
google him
I get chess master
@leslietownes im dying lol
8:02 AM
Yeah i got the point, he's amazing...
the tsetlin stuff is even on his main page, i'd never heard of that
he's maybe one of the best mathematicians of the 20th century
i never met him
met a few of his students
Anyways, I need to study for my econs exam. Bye-Bye!
:) glhf
I leave with you a brat. Take care of him.
i refuse to sign
8:06 AM
that's a great costume
looks like the kid, like my daughter, has not had a haircut since covid-19
its 1 in the morning, my cat just got up from her bed downstairs, came upstairs, meowed at me, rubbed my leg, and then went back downstairs
cats are inscrutable
needed to check if food bowl will be refilled tomorrow
my wife does the morning bowl
she [the cat] flips out when i'm on conference calls because she can't tell who i'm talking to
whats the solution lol or do you just let everyone hear the meows
or scratching etc
8:14 AM
the latter
and god help me if it's zoom
some of my teams know her they say "hi olivia" and i'm like DON'T ENCOURAGE THIS
she aint need no encouragement, its for the love of flipping out
i've had a few meetings where she's right in front of the camera so if you do gallery view it's like a cat is having a conference with three other people
lol there was a person who turned on some cat filter right
and couldnt turn it off
8:17 AM
yes at a court hearing
"i'm not a cat"
i felt for that guy
i was once on a phone teleconference with a judge and the guy who was doing the talking turned to me and hit a button on the phone that should have been 'mute' but was, in fact, 'hang up'
first hit on youtube for im not a cat hahahaha
and we looked at each other and were like "uh oh"
ahh so good to laugh at others' misfortune (sorry everyone)
technical errors must be all over the place still
8:22 AM
the worst thing about eating shit as a lawyer is there is a court reporter literally writing all of it down
and transcripts are publicly available
ooh thats double bad
one time i had to appear at a hearing that we knew we were going to lose, and lost, and in the time it took me to drive back to the office they had received the transcript
have they been writing every single word tho. like even at the start of the call, can you hear me etc
so i just said "yup" and went back to my office
reporters do put a lot of that preliminary stuff in, which is why i don't say any preliminary stuff
i expected them to start when the judge gives a signal or smth
8:25 AM
you see this even at the US supreme court, sometimes the official transcript includes stuff 20-30 seconds before the actual argument begins
has it ever helped or hurt a case? to your knowledge
no it makes no difference to the case, it's just lulz for people who know that it's there if it's funny
haha ok
i guess i could forsee especially with zoom that someone doesnt mute at the start and says something they shouldnt say which then goes into record
in the transcript of the last decision i lost, which was four years ago, we knew we were going to lose and were going to appeal so i told the judge at the outset, we'll go on the briefs, and there was argument from opposing counsel guided by the judge, leading to the judge saying something, and the judge asked me what i had to say in response, and i said "we disagree with absolutely everything you just said"
and he kindly told me to f off and we had a nice transcript at that point
cool way to make a living
we settled before the appeal so everybody won
math.stackexchange.com/questions/4455409/… is probably a dupe but a good exercise in group theory
$|G| \mid |H|$ is technically correct typesetting but I hate it
9:02 AM
hola chat
A quick sharing: I'm happy
when learning linear algebra
good for u m8
thank you :) Calvin
linear algebra is cool
9:05 AM
3blue1brown makes it accessible to normal people like me
before there was 3b1b there was Halmos
imo a sicc book
Guess what, I have a (privated?!) pdf lol
did try to read it before, but it's still too hard for me
i meant pirated
oh lol
I'm private pirate living in a ... TBD
shh theres a lawyer literally right there
9:08 AM
Never mind, it has already been deleted since I cannot understand it, lol
I was too poor to buy a new book when I was a student
so I can only searched around free books online (and I never further shared it to any others)
I recommend watching this one, my hero: youtu.be/R6S-b_k-ZKY
k looks fun will watch when i cant be bothered to compute more integrals
integrals are like girls, at first look they're beautiful
At second look, it's complicated and (fill in your answer here)
be careful not to get cancelled
thank you Calvin, you saved my afternoon, I'm leaving
9:30 AM
how did i do that
you're welcome tho
9:57 AM
@DiagramChasingBlues :-)
1 hour later…
11:08 AM
hola chat
waiting for my pizza now
Saturday, pizza day
@linear_combinatori_probabi do you speak fluent Spanish?
@bumblebee I do, how can I help you
I have my Spanish exam soon, could you help me practice.
they teach you spanish in singapore? thats sick
International school tho
11:24 AM
yeah duh ofc gelfand's friend's granddaughter goes to an international school
I have this question: Escribe entre 60-75 palabras sobre qué tiempo hace hoy donde vives y qué tiempo te gusta o detestas. Debes utilizar estes palabras: 1. Me encanta 2. El mal tiempo 3. Prefiero 4. El paraguas.
This is my translation: Write between 60-75 words about the time where you live and the time that you like or hate. Use the words: 1. I love 2. A bad time 3. I prefer 4. The umbrella.
Is this correct?
@CalvinKhor U went to an international school urself.
:) ill leave you to your spanish. gl m8
So @LeakyNun is it good?
@bumblebee (hace) tiempo = weather
That was a test to see if you actually knew Spanish. Sorry, i've been trolled too many times.
Now let me type out my real question
11:29 AM
Could you check that I conjugated everything correctly.
and written the correct words for the things in bold @LeakyNun
Gotta eat dinner now
¡Hasta luego!
@bumblebee e and f should be past tense, h should be past tense (not past participle)
12:10 PM
Thanks Leaky
12:39 PM
Jeez, some people on mathstack are just not it. My post is getting downvotes. Now people are upvoting the comments saying the question is confusing even after I clarified the commenters doubt. I ask users on CURED to help me reopen the question. They call it "such an off-topic question". But when I ask for reason and justification they ignore. Honestly, i'm thinking of leaving mse, this site is so biased and some people here are just not it.
12:51 PM
1:08 PM
@bumblebee Write "bumblebee left mse" in your username and we can be best friends :D
1:19 PM
when did desmos turn blue?
How can I show $so(3)$ and $sl_2(\Bbb R)$ are not isomorphic as $\Bbb R$ Lie algebra?
Using generator is nasty. Is there any better way?
someone downvoted a post of mine without giving any reason whatsoever.
1:46 PM
@Prithubiswasleftmse I haven't reached that point yet.
But I am close.
@bumblebee you will get there eventually.
Soon we will have an entire army of "left mse".
I have my pride. I'm not gonna waste my precious time on people like them.
yay another downvote for no reason! im loving it!
Hello, i am trying to prove the grassman identity using the language of vector analysis and differential forms ( IE No BAC CAB no vectors and diffs etc, just translating the vector fields into one forms and then working it out) however i am not sure how you would understand the laplacian of a vector field in this language?
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