Mathematics - 2021-04-23 (page 1 of 3)

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12:16 AM

@AndrewMicallef apology accepted, you may go about your business. but be sure not to recidivate, for the line is growing ever thinner

the line is also the result of a conformal map, guess which

12:27 AM

Here's something that I think I saw on a competition a while back but cannot remember the source: suppose $a_n$ and $b_n$ are sequences of integers so that $a_n=b_n$ for all large enough $n$, $\sum_{i=0}^n a_i=\sum_{i=0}^n b_i$ for all large enough $n$, and $\sum_{i=0}^n\sum_{j=0}^i a_j = \sum_{i=0}^n\sum_{j=0}^i b_j$ for all large enough $n$. Prove that $a_n=b_n$ for all $n$. Does this ring a bell for anyone?

12:42 AM

gosh, what a problem. no bells are ringing here.

12:54 AM

yeah

I don't know

i was thinking about just grabbing a minimal n for which the hypothesis isn't true and a kind of induction. i can't think of how to use the fact that they're sequences of integers.

this is why i never competed in competitions.

any idea on my question?

If a straight line starts at $(0,0,0)$ and ends at $(1,1,1)$ then projecting this line onto each face of the cube will yield a line that bisects the face.

i'm not sure that i know what you mean by vector field or what the end game is. you can construct all kinds of vector fields on (0,1)^3 but do you want something interesting or special to happen at the corners (which as defined it does not have)? do you want it to match given data on certain surfaces? ??

Does anyone know here of a (simple) example of two independent random variables $X$ and $Y$ such that $X+Y$ and $X-Y$ are not independent?

@leslietownes I want the vector field in (0,1)^3 to project onto each face and return a prescribed vector field on each face

1:00 AM

"A classic result in probability theory states that two independent real-valued random variables having independent sum and difference are either constant or normally distributed with the same variance."
Oh, apparently anything not equal to a constant or normal distribution should work

i was about to say

@leslietownes so there is some "decoration" on each face of a cube and the question is how to re-construct a possibly unique vector field in the interior which projects orthogonally onto the faces, returning the "decoration"

i think you might be asking one vector field to be doing too many things. six things is a lot of things for a poor three-dimensional vector field.

no rigor just vibes

yes it's a strong restriction on the vector field

meaning there could be just one unique vector field that satisfies the criteria

something like independent coin flips ought to do it.

i miss probability theory.

1:06 AM

@leslietownes Ye, I did that one!

I guess I'll ask cause nobody knows

great minds think alike.

At first I was a bit intimidated, since I thought I had to come up with a fancy example, but as soon as I found that thm, I reckoned, let's do the easiest one then:)

@leslietownes x'D

yeah i was thinking for a minute i'd have to write out integrals or something.

haha

geocalc just saw it posted on SE. is there a motivation behind this question?

beyond what i can infer from the question itself?

you could do some kind of convex combination thing where you weight by how close you are to a given face. this might be difficult at the boundary. and it seems highly non-unique because you can play with how you weight things as long as they approach the right thing when you go to the boundary.

yes there is motivation. It could give a recipe for building 3-vector fields from 2-vector fields

1:18 AM

put a zero on the end? :)

do you require these fields to have any geometric properties? do they have to be tangent to a submanifold?

the 2d ones?

i'm just wondering what 'vector field' carries with it beyond function into R^2 or R^3

yeah I could require them to be congruent to $X=(x\log x, -y\log y)$

congruent meaning what, exactly? i'm not a geometer. talk to me like i'm dumb, because i am.

congruent meaning they are the same but configured in 3 space at different locations

so if you leverage the euclidean transformations you can map all 6 on top of each other and they will be indistinguishable

1:23 AM

shock g of digital underground apparently passed away. this may not be big news for people who haven't spent a good deal of their formative years in oakland california.

shock g

cinema fans will remember his performance in dan aykroyd's 1991 directorial debut "nothing but trouble." 2pac is visible.

I must be boring you

oh, i'm always bored, but not by you. i'm just trying to conceptualize the question. it seems overdetermined and underdetermined at the same time.

it was 'ladybug day' at my daughter's day care. she came home with a black-spotted red paper crown with eyes and a smile on it and pipe cleaner antennae.

nice

that's cute

1:31 AM

user image

that's a nice thing that happened today.

wow 😊

we should do ladybug day at the office.

yeah you should suggest that

hi office manager, i propose that casual fridays be replaced with ladybug day. attached is an example of the headgear that i think office services could create for all of us. it would increase workplace cohesion and probably help to retain new employees.

we have all of the relevant material. we may have to put in an order for googly eyes but we have the other stuff, including pipe cleaners.

why we have that in our office services department is a mystery to me, but we have them.

hope they accept your proposal

1:36 AM

i want to attend a zoom hearing with a federal judge on ladybug day.

That'll be the end of this career.

i reinvent myself in ten-year cycles, almost like the cicada. i think in my next career i will be a motivational speaker.

Do you have what it takes?

I think every motivational speaker could use a ladybug headdress.

Steve Martin made it big with balloon animal headdresses

1:59 AM

when you don the ladybug headdress, it gives you an undeniable edge on the competition. this will be central to the motivational program.

Andrew Micallef

2:41 AM

Mwha ha ha circle paramaterized, and I got the right answer!

Can other beetles be welcomed on ladybug day?

Mostly cockroaches.

Andrew Micallef

Ahh spoke to soon, wrong answer

Probably an algebra error lurking somwhere

I was devestated as a kid to find out there were male ladybugs

I thought I got it right the first time, plus my break is over: therefore problem solved, because I think it therefore it is. QED

2:56 AM

my cat is still looking for another cockroach to play with.

iowa had some invasive species that looked like a ladybug but wasn't. in the fall they'd crawl all up the side of the math building, or any sunny surface. gobs of them. it was pretty gross.

Maybe the ladybug headdress will placate her.

my daughter did attempt to put it on the cat. the cat was not interested.

@AndrewMicallef the circle in Ted's book?

3:39 AM

@robjohn Ayup

188 188 188 76 34 44 44 146 194 204 44 178 106 114 210

uh oh, looks like the u-boats are after our shipping again.

Anyone who decodes my message will be considered a true cryptographer

4:01 AM

it doesn't seem to be ASCII. that's all i can say.

my thought process was, is that ASCII? then, 188 is a pretty weird one. and three of them? unlikely.

anyone who wants to hire me as a cryptographer is welcome to do so. starting bid is $500,000 a year and i don't have to go into any office.

4:17 AM

i meant $750k per year.

i'm going to need one million dollars. annually.

qqq m g hh k n f h p g f c

is that right?

i'm here for my $5 million contract, no other reasons.

$1 billion annually. Take it or leave it.

@geocalc33 no

4:34 AM

I added the digits of each number

then mapped them to letters

Nay thy method is wrong

I'm just showing my method

Maybe I won
Seems like nobody can decode my message

4:50 AM

i've decoded it and sold it to the russians

Blin

y did u do dat

they offered me very generous financial terms and a fun party in a forest.

wait why is the russian military banging at the door I didn't do -

that was part of the terms and conditions. i apologize but i'm not sorry.

any activity in a moscow hotel room?

4:59 AM

no i was mostly in st petersburg.

user image

this is probably the best encryption system i have ever devised

chat cannot show some of the characters so i had to upload an image

I like this: incoherency.co.uk/chess-steg

i encrypt my data in a background signal in a zoom call.

5:16 AM

scribie.com/files/3b840a5489a6481698424d017886f17332eb1fa1

do you want my swiss bank account info as well?

actually, it really is an offshore account

no i actually don't need it

what shall i doth with thy account info i am not a criminal

few announce their criminal status.

Gêse yfel becnâwan ðone as

@Euler2 can you improvise messages or must you use a computer program to generate that?

5:22 AM

Yfel ðênung dôð latost

scribie.com/files/9e5d869cd3514afa849443205f5fbf30f2a5f6a5

1 hour later…

6:49 AM

@copper.hat no

anyone like music

@geocalc33 yes one of my favorites is this: youtu.be/dQw4w9WgXcQ

good song

Yeah a masterpiece

Everyone should listen to it

soundcloud.app.goo.gl/AcVUXSsVHwgk5U1s9

here's a song

does the link work?

If $\exp: \Bbb R^2 \to \Bbb R^2_{+}$ how does a series transfer under this map?

for example, if $\sum_{n \ge 1} \frac{1}{n^2}$ is defined in $\Bbb R^2$ how does this transfer under the map?

The sum in question is 1+1/4+....

2 hours later…

8:42 AM

nvm figured it out

9:05 AM

Hello Everyone! Hope you are safe. I have a research paper (which comes under the Number Theory subject class) that I would like to submit to the website arXiv.org whilst it is in the process of being peer-reviewed (which I realise can take a very long time). However, the submission guidelines say that it must be endorsed by someone currently affiliated with arXiv.

Anyone from Math stack exchange community would like to endorse me on arXiv? As said on the the arXiv website, I have tried contacting such verified people but they never respond to my emails. I could've asked my mentor for the …

9:37 AM

I've substantially edited this question. I would love it to get a bit more attention, as it really seems like something that must have been studied before, but it seems hard to Google. math.stackexchange.com/questions/4112437/…

Andrew Micallef

10:00 AM

@robjohn yeah the circle in Ted's book (also I totally did solve it, when I thought I did, I did just screw up the algebra, and I only know that because of wolframalpha, not because I worked through the algebra)

user image

I got an expression for t in terms of x and y as illustrated above (using trig) then substituted my x and y into the equation of a circle $x^2 + y^2 = 1$, then had a headache solving for x and y in terms of t

...but you all probably knew all that already

which book is this @AndrewMicallef

I'm diving into Analysis I by Tso at the moment.

sorry, I think your book is a different author.

I was thinking it was the same author for a second.

epsilon-emperor

10:35 AM

5

A: Find polynomials $p_n$ on $[0,1]$ with integral $=3$ and converges pointwisely to $0$

Take $p_n(x) = 3x^n(1-x)(n^2+3n+2)$. Each $\displaystyle\int_0^1p_n = 3$ and $\displaystyle\lim_{n \to \infty} p_n(x) = 0$ for $x \in [0,1]$.

Could someone please explain the motivation behind the accepted answer here?

It works, sure, but HOW DID the answerer think of it?

Andrew Micallef

@zacts Ted's book on differential geometry

thanks

Andrew Micallef

anytime (I was really only sharing because after drawing the friggin diagram I felt my eveing would have been a total waste if no one ever saw it)

I do know of a non-traditional book on differential geometry. (Note: I haven't studied the subject at all yet, and I'm just getting started with mathematics). With that note in mind here is the link: mitpress.mit.edu/books/functional-differential-geometry. It uses the programming language scheme to approach teaching the subject.

The PDF is free creative commons. I don't know if it may or may not be useful or interesting.

@AndrewMicallef that was for you by the way.

I'm currently constructing the integers via the peano axioms in Tso's book on Analysis.

I'm hoping the exercises and problems work out for me.

11:27 AM

I really like Tso's writing style in this text.

11:44 AM

Sorry, I meant Tao's book on Analysis. I'm getting way way too sleepy to chat. I'll be back later.

11:59 AM

@epsilon-emperor Look at Don Antonio's answer. He found a sequence of polynomials that tend to $0$ on $(0,1]$, but does not do so uniformly, so it does not tend to $0$ at $0$ (that was the problem with his answer). That is the right idea, but misses the convergence to $0$ at one point.

The accepted answer flips the domain and uses $x^n$ instead of $(1-x)^n$. This has the same problem at $1$, so they multiplied by $(1-x)$. This gives $x^n(1-x)$ which DOES tend to $0$ over all $[0,1]$. Then they divided by the integral over $[0,1]$ and multiplied by $3$

$\int_0^1x^n(1-x)\,\mathrm{d}x=\frac1{n+1}-\frac1{n+2}=\frac1{n^2+3n+2}$

so $p_n(x)=3\!\left(n^2+3n+2\right)x^n(1-x)$

Can you show that $\sum\limits_{n=0}^\infty p_n(x)=\frac6{(1-x)^2}$ on $[0,1)$? Why does that and $p_n(1)=0$ prove that $p_n(x)\to0$ pointwise?

12:16 PM

What is a sequence that is bounded away from zero but neither positively nor negatively called?

I don't know. What is a sequence that is bounded away from zero but neither positively nor negatively called?

Sorry, that misses the point ;-) It sounded too much like a riddle; I had to. But honestly, I don't know what that would be called.

it is definitely non-convergent, to any value (assuming there are infinitely many positive and negative values)

12:35 PM

i think "bounded away from zero" is perfectly fine terminology

12:49 PM

it is, but I think they were looking for a term that more specifically meant "on both sides".

Yes, for example a sequence like 1,-1,2,-2,...

I guess I'm not sure what's actually being asked for, then

"bounded away from zero" should mean "not having zero as accumulation point"

1 hour later…

2:13 PM

i wonder if the two-sidedness is important. i.e. infinitely many n for which x_n is positive, and same for x_n negative. for "both sides."

2:26 PM

Two things, 1. Does anyone see a problem with my working on why a convolution with a Lipschitz function is Lipschitz math.stackexchange.com/questions/4111300/… and 2. can I confirm the notation $W^{1,1}_{loc}$ means functions (and their 1st weak derivatives) which are $L^1$ integrable over any compact set?

i don't know what the wasserstein metric is. i assume it's not named after the investment banker.

that looks like an OK definition of W^1,1 loc to me

@leslietownes thanks

they're often called sobolev spaces. i don't know why W is used, if we're supposed to be paying tribute to sobolev.

his real name was Wobolev

wasserstein has a cool web page. personal.psu.edu/lxv1 very late 90s vibe.

2:34 PM

that is a cool photograph

if you have a metric named after you, you get to wear sunglasses indoors.

I have many things named after me

i still wonder what it's like to have the last name 'euler' and be in a math class. there must be no end to the abuse from your fellow students.

Change it to something like wheeler or youler

Wholer?

2:39 PM

Yes

oh, Youler.

Youler version 2.0.0

this reminds me of a guy who lowkey tried to leverage the fact that a result in his field was authored by a guy with the same last name, to confuse people into thinking maybe it was his result. he never directly claimed it, but went right up to the line.

klever

it was pretty shameful behavior, if you ask me, but on the other hand, anybody who was fooled by it was probably an idiot anyway.

works in finance now. that seems about right.

2:46 PM

thatsmathematics.com/blog/wp-content/uploads/2012/09/…

This is a randomly generated paper by mathjen

The interesting thing is, it was accepted by a journal

Although he had to revise a bit

But

user image

5 mins ago, by Euler 2

This is a randomly generated paper by mathjen

*mathgen

If I have a potential $\nabla f$, and I want to show it is in $L^1_{loc}(\mathbb{R}^d)$ ( i.e each component is in $L^1_{loc}(\mathbb{R}^d)$ ) what is a relaxed assumption I could put ( say on the hessian of $f$ )

3:04 PM

mm, not too surprising that an automatically generated paper could get published somewhere. not to be elitist (i never published in great journals) but a lot of journals are very much pay-to-play and it is mostly about dollars and euros landing on accounts somewhere. $500 "processing charge" indeed.

speaking of that, the Leslie Townes Journal of Pure and Applied Mathematics is accepting submissions.

hah, i just noticed that the references are also randomly generated.

i would have left that out and used randomly chosen real references, but their way of doing it is funnier.

if anyone has a copy of V. E. Euclid's "Commutative Number Theory" i would be interested in purchasing it

3:38 PM

@AndrewMicallef That diagram is mislabeled. The length $x-(-1)$ is not what you have marked at all. And you complained about my $t$ next to an arrow. ... The algebra is quite easy if you notice some factoring that can be done with the right arrangement.

@leslietownes and that says what about finance people?

nothing nice, i guess. can't really talk, lots of goofballs in my industry too.

@leslietownes my wife is an attorney, so I would never say anything deprecative about those ~~sharks~~ folks.

i had a surprisingly productive phone call with opposing counsel yesterday. i had cleared out about an hour for it, but it was over in 7 minutes. imagine if the whole legal industry worked like that.

it might cut into my pay. let's keep everything the way it is. my next call should be 2 hours to balance it out.

4:02 PM

Sloppy math.

Besides, don't attorneys charge 1/2 hour for 7 minutes?

we bill in tenths of an hour, so it would be 0.2. of course we round up.

never round down.

Oh, a family law attorney years ago billed me a few hundred for literally a 5-mInute consult.

that happened to someone, it might have been chern? if he had a divorce. wu recommended somebody an attorney, and they had a productive call on the order of 4 minutes, then this enormous bill arrived.

we don't do that.

i mean we do enormous bills, but not surprise ones. there's usually an agreement in place first.

Not Chern. Dedicated to his wife throughout.

oh, i remember who it was.

4:08 PM

For me it was before I hired an attorney in ATL to get guardianship of my Alzheimers-plagued mother. Such fun.

alzheimers is horrible. we had a big thing about that when my grandmother was in hospice. i'm glad my practice is not adjacent to anything like that.

everything i'm on it's mostly people yelling at each other about other people's money. investor money. investors aren't on the call. it keeps things gentlemanly.

4:43 PM

You recently interacted with a user on Math SE who claims to be 12 years old. Please note that users are required to be 13 or older in order to register for an account here. If you encounter a user who claims to be under the age of 13, please help us out and raise a flag.

Regards,
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waiting for the police to arrive.

oh goodness

math.stackexchange.com/questions/4112655/a-student-says-a-b-b-a-is-this-statement-always-true-sometimes-true-or-never?noredirect=1#comment8505804_4112655

when i looked at that last evening it appeared that the most interesting parts of the dialogue had already been deleted.

to be fair, albeit i removed it quickly, my first response was something along the lines of 'ffs, do some work'.

so far, the people i have met from the internet and my underage dealings have both been on mse.

the moderation seemed excessive to me, although i was raised by wild animals and do not have a sense of what is or is not appropriate. ffs would not have offended me at 12 although i think i would have understood its intended meaning.

4:47 PM

expecting mse police with fixed point guns & quotient ring flak jackets to show up

yes, enjoy your interaction with MSE paramilitaries. you'll be used to it from your childhood in ireland.

if i thought i was dealing with a 12yo i certainly would not have responded as i did

:-) a breeze in comparison

How did we establish the OP’s age?

there was a comment where he said what grade he was in. his grade may have been confused with his age for a minute.

it's all fake news to me. if anyone asks, i am 29.

Yes, Jack Benny.

4:49 PM

i am 60 in earth years apparently

mentally, well...

jack benny is one of my strongest influences.

i also have a full head of hair.

there was one about boxers and ballet dancers being similar...

my nose is smaller than bob hope's.

also i have twenty zillion dollars.

oops, kettle's boiled. my late day is starting

that helps as long as zillion translates

i'm 64 oz into today's herbal tea binge. mostly turmeric ginger.

4:51 PM

no matter what i do my financial worth is analytic and bounded.

and not very large at the origin, unfortunately

you need nevanlinna theory to quantify my net worth.

i will pick up on that

i sleep poorly but last night i slept comparatively well (2x 4hr sessions) and i feel completely out of it now.

how strange.

the day after i got the first corona vaccine i slept like i hadn't slept in years. they should prescribe it for sleep disorders.

the first just left me with a slight soreness 6" below the injection site

i have had rabies shots that has less impact

my whole arm was tingly and numb for about 36 hours and i had a bad fever. #2 is supposed to be worse.

4:55 PM

wow. sry to hear it. my 'boss' was knocked out yesterday as a result.

a guy at my work said the first pfizer could be worse if you'd had covid already. i may have had it last fall. i had about a week where i couldn't taste much and spent most of my time in bed.

eating was such a chore. i didn't want to eat because i couldn't taste anything, but i knew i kind of needed to eat. it was a pain in the neck.

uuugh

but never in the depths of my illness did i type "ffs" in a comment section and arouse the wrath of internet authorities.

shame on you.

deep down i am a logical rational person. but the little red guy with the tail on my shoulder makes me do things

it's an abbreviation for "For friendly supplementation." i don't understand why people are overreacting.

what do they think it means?

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