Mathematics - 2021-07-27 (page 1 of 2)

copper.hat

00:39

its astounding that there are no comments on the video.

Yorch

they are waiting for your comment

copper.hat

isn't stanford part of berkeley, so i guess including sri is ok...

Yorch

I like when their master tellls them they can stop cycling

copper.hat

@Yorch i am sure they got dragged into it against their will.

in my life experience the power flow is in the opposite direction

Yorch

what do you mean ?

copper.hat

00:43

are you married?

that was rhetorical

Yorch

no

but there's two people in the bikes

and the person filming tells them they can stop

copper.hat

i did figure that part out, thanks :-)

Yorch

Cool

geocalc33

it's not the density of points in the plane, it's the density of plane in the points

Yorch

I still don't know what you're saying

Is the working theory that the camerawoman is the wife of the dude on the bike or something?

copper.hat

01:00

it was a broad observation on life, you introduced the term master, i was suggesting that it may not necessarily apply.

the point is the plane density

Yorch

01:11

Oh I see

leslie townes

i understood the SRI to be the silly valley one and not the berkeley one but could have been wrong.

stanford is berkeley's ugly stepchild.

robjohn

01:36

@TedShifrin SRI is listed in Berkeley, so I thought it was a precursor, but maybe not

LSS

Guys

Hello

Someone could help me?

I am having trouble to understand how to find the boundary condition for the continuity equation divJ + d\rho/dt = 0

Should J be continuous at the boundary?

robjohn

@leslietownes You're right; SRI is the Stanford Research Institute.

leslie townes

boo. stanford sucks.

it's amazing now but in the 1950s-60s it was summer camp for rich people. although noted president herbert hoover did go there and did his best with a bad situation.

for a while at berkeley i worked at something called the electronics research laboratory, which is a little weird because i don't think i did anything relating to electronics.

robjohn

However, the first ARPAnet communication went from UCLA to SRI

it was a logon

leslie townes

that's cool.

early internet stuff is very cool and if it didn't go to the basement of evans hall it is still OK with me.

even if it was, ugh, UCLA.

i do law recruiting there and the students are very sharp. and the math department speaks for itself.

copper.hat

01:58

erl. that paid my way there.

i think my fave on campus place was the math library.

i miss humphrey gobart

robjohn

@LSS More context might be needed.

leslie townes

02:16

you are also an ERL alum? what a small world.

copper.hat

yep. i really liked hardware but saw none of it while there. the irony of it all.

i mean hands on stuff

leslie townes

i was designing an algorithm and we ended up giving up. electronics were not expressly involved.

i think they were in cory although the grant administrator for my project was in soda.

russian bot

02:36

meinkraft

copper.hat

03:05

most of my time was in cory, then evans and i avoided rechtel if i could

i spend a good bit of time in the physics basements

leslie townes

03:23

all of those places have been constantly slated for demolition for seismic reasons for 20 years and yet they have no money to replace them with anything.

they may have done campbell. that's where math was before evans.

TheBestMagician

04:03

hello

leslie townes

good evening.

copper.hat

04:49

hi

robjohn

yo

leslie townes

the suspense is killing me.

copper.hat

drum roll...

Edward Evans

Yo

copper.hat

expecting half a pretty female assistant to appear

i guess the best was hyperbole. back to simplifying my incomprehensible home it setup.

leslie townes

04:53

always helpful to simplify the cabling.

i went off on a coworker once, not as you jokingly did but because as he unacceptably was treating a female coworker as a pretty female assistant. i was shouting. it is legendary around the office.

i just lit up. it's nice that my coworkers think it was ultimately funny. genuine behavior in that area, which is what that was, is pretty gross.

at home i joke more or less all the time about being a sexist pig.

my wife understands that the horrible things i say are a product of my environment and not who i really am.

robjohn

@leslietownes and now she thinks she isn't pretty ;-)

leslie townes

oof, probably. i hadn't thought of that.

robjohn

Hey, @Ted!

leslie townes

at home i say the worst things i can think of because i think they're funny, but if i hear them at work, it's game over.

Ted Shifrin

Oops ... Didn't realize I was still logged in here. Hi, @robjohn et al.

robjohn

05:00

you just floated in

leslie townes

good evening mr. shifrin. we have your usual place at the window.

2

Ted Shifrin

I was at duplicate bridge, but just came home and woke up the computer, yes.

Edward Evans

Hullo @Ted :)

Ted Shifrin

Heya Edward. Stranger.

Edward Evans

Exams are over, I'm back to haunt

lol

robjohn

05:02

duplicate bridge; is that in case the original bridge fails?

You might need a contract to replace it

leslie townes

i've never played bridge. my grandmother was very into whist and the approximately 90 minutes of my life i spent with her were learning that game.

Ted Shifrin

Bridge is a challenging game ... helps my brain not be totally dead :P

My grandparents played canasta, no bridge.

leslie townes

it seems to be very difficult.

oh, my grandmother loved canasta.

robjohn

My parents had numerous bridge parties at their house when I was a kid. Pretty well guaranteed that I never played it.

leslie townes

my dad taught me a cool card game. it goes by the name of 52-pickup.

Ted Shifrin

05:06

I played pickup stix.

leslie townes

it was so funny.

Ted Shifrin

My parents played casual bridge, but I only started learning in college, and then took it seriously only in my adult life.

leslie townes

there was a lot of it in 1015 evans when i was in grad school but i never understood it.

Ted Shifrin

There was go in my day, not bridge.

leslie townes

there was some go, too. that is far too intricate of a game for me to be involved in.

it's really f---ing hard. pardon my french. in chess you can sometimes get over on someone with dumb tricks.

my one-year college roommate grew up in the soviet system and fancied himself a master at chess. he could only draw against me.

way too aggressive.

i never beat him but we drew all the time.

he'd always get into something. especially when playing white. he couldn't help himself.

chess is a psychological game.

Koro

05:14

Hi Leslie, did you every play chess with an online bot named Boris? :)

leslie townes

i don't think i have. i have done some blitz chess with mixed results but i prefer the full game.

my playing style is slow and boring and just setting up the position with the most boring things imaginable.

it sometimes works but i often lose.

Koro

:) i used to play chess a lot, i was very inspired by style of Mikael Tal.

leslie townes

he was a complete inspiration.

i like viktor korchnoi.

Koro

the way he played was fantastic :)

leslie townes

korchnoi was great more or less forever, until he died. that is an inspiration to me.

and he would kill people by playing passively and then sliding in the knife.

Koro

05:19

i never heard about korchnoi :-(

leslie townes

you should look him up. he almost beat karpov but did not. he would kill people at seniors tournaments.

Koro

yeah i'll look

leslie townes

he was alarmingly consistent in the quality of his playing until he died at age 85. we should all be so lucky.

copper.hat

i like magnus

leslie townes

he's amazing.

i don't even have words for it

copper.hat

05:23

he sortof is an all rounder

he has stage presence, seems like a decent human

Koro

i have seen some of his videos where some challengers were found using engine and still they lost :)

leslie townes

yeah, haha. good luck.

i'd put my money on magnus.

copper.hat

for me chess is like table tennis & math, there is a level i play at and that more or less is it. depressing at some level until you accept it

leslie townes

copper that's exactly it for me.

i'm happy if i can beat people in my family, but i suck.

copper.hat

i improve by lateral moves :-)

in life

Koro

05:26

it would be fun to see magnus vs stockfish 7 (is 7 the highest level of stockfish?)

leslie townes

and i can't win, i can only draw.

i've drawn against an IM.

enough for me.

copper.hat

i can grab defeat from the jaws of victory, i am creative but no memory for closings

leslie townes

a lot of the closing game does come down to memorization.

copper.hat

and openings, but i am usually more awake then :-)

somehow the olympics just not grabbing my interest

leslie townes

when i play i try to get out of the usual openings as quickly as possible just to see what the other person can do.

the olympics are boring me.

copper.hat

05:29

to combat my memory i like 3+0 blitz

leslie townes

that's brutal but of course you would like it.

copper.hat

i'm amazed the world standardized on time but nothing else

i can drink wine and blitz :-)

no deep category theory thinking going on

leslie townes

we put out a cricket trap in our garage because there were too many crickets in there and then a lizard got into the trap and was immobilized by the adhesive and died. i feel really bad about that.

copper.hat

i don't like when things die at my hands, but what can you do

leslie townes

i did want the crickets to die.

i didn't know we had lizards in the garage, at all. i've seen them outside but not there.

copper.hat

05:32

a little bird fell out of its nest prematurely

i leave it there.

my friend didn't want it to suffer so he crushed it with his shoe

leslie townes

80% of birds die more or less immediately.

i love birds but it's a fact

copper.hat

i know, but i'm still disturbed by what my friend did

we have done much worse, but this one was on the boundary of what i can handle

leslie townes

it is wrong to crush anything.

copper.hat

that would be my general option with one exception

earwigs

leslie townes

if something might be in pain, you kill it immediately. but you wouldn't crush an animal.

that seems wrong.

i don't know.

copper.hat

05:34

he was quick about it, he did not want to inflict pain

leslie townes

in his defense there is very little chance of getting back to the nest if you fell out too soon.

he might have just put it back in the nest, though. it's an urban myth that birds will not care for a bird that has been handled by humans.

copper.hat

this was the case here. somehow years of catholic upbringing lets me believe some winged thing will rescue it :-)

the nest was out of reach

i would have taken that risk

leslie townes

well, sh-t.

the animal was not infected with original sin and is likely in bird heaven now.

copper.hat

i'm going to ask tarantino to make a movie about it, pulp friction

that was bad, even for me

leslie townes

but very much in character.

copper.hat

05:37

the female volley ball teams are playing, a bit distracting

not really a volley ball fan

leslie townes

oh god, i went over this with my bff.

i don't understand the sport but do like people in skimpy outfits. she does too. i think that's the only reason it's an olympic sport.

just a bunch of people perving out about it

copper.hat

norway disagrees

the team refused to play in bikini bottoms.

capital offence i say.

at risk of misinterpretation, generally i think the human form in great athletic shape is something to behold.

leslie townes

i'm generally against the olympics because while there are a tiny number of rags-to-riches type inspirational stories it is mostly celebrating people who had an enormous number of resources growing up.

it seems like a misallocation of attention.

i'm still watching girls volleyball.

copper.hat

i knew a few hopefuls and have met some irish olympians (runners & boxers)

none had resources

i have met a few us olympians

one was out neightbour at some stage

he had the distinction of being both a summer & winter olympian

leslie townes

that's weird.

copper.hat

05:42

another taught my kids to swim

i never had any concern around water as a result

he was a totally nice humble guy

leslie townes

the only olympian i knew was born rich and had her parents put literally all of their time into her sports career. it seemed excessive, but that is the vibe i get from a lot of them, at least from the USA. too many resources poured into a very weird box.

copper.hat

i only found out when we went out for lunch (which we often did) and he was wearning a leather jacket with the olympic circles on it

"where did you steal that dave?" asked joe

leslie townes

that's 100% what i'd say.

copper.hat

"i was a summer & winter olympian joe"

leslie townes

we might be the same person.

copper.hat

05:44

" nfw dave"

i had to eat a large crow enchilada pie

i mean he could at least have had some unlikable characteristic so i could feel good

leslie townes

one time i was at a legal event, and i was talking to someone whom i did not know was a federal judge who said he biked there, and i said "oh, DUI?"

copper.hat

:-)

leslie townes

he has since been on one of my cases.

Edward Evans

@leslietownes The UK gold medallist for synchronized diving of some description was in my year at secondary school

copper.hat

i have met a few people in that manner with my lose cannon mouth

Edward Evans

05:46

not such a great guy. I commend him for his achievements though

leslie townes

diving is a dumb sport. i said it.

Edward Evans

it's only dumb if there's no water under the diving board

copper.hat

my sister's running club is run by an irish boxing medallist

leslie townes

i can splash in the water. give me a few weekends and i'll be in the olympics.

copper.hat

i haven't swum in 1.5 yrs

swum swam?

leslie townes

05:47

you just point at the water and go.

it's stupid.

Edward Evans

The diver made some claims about having been bullied constantly, despite being wildly popular, and was moved from our state school to a massively over-priced private school on a scholarship

copper.hat

i love playing in the water, preferrably with waves and not freezing

leslie townes

no disrespect to your friend, edward.

Edward Evans

He's a bellend

rofl

leslie townes

or acquaintance.

copper.hat

05:48

a glans of a man?

leslie townes

bellend is one of my favorite terms in the english language.

Edward Evans

A glans indeed

copper.hat

it is a problem when one comes up with a great sentence but you know your presnt company will just not get it

in fact they will think it is more than strange

leslie townes

my wife did sports a little bit in college and every one she was around was a toxic and horrible person. just really horrible people.

Edward Evans

ll

lo*

leslie townes

05:49

i spend my life trying to be more than strange.

Edward Evans

wtf

copper.hat

odd. generally i found the athletic/sports folks to be decent off the field/mat

even the two olympic judo hopefuls who lived off the dole to support their sport

leslie townes

the athletic men i have known have been ok but apparently if you're a girl and you work with girls it can get very awful very quickly.

copper.hat

would not want to meet them in a dark alley...

Edward Evans

Maybe it's being in a solo sport or smth

leslie townes

05:51

my wife has a black belt in karate and was disqualified from a tournament for breaking someone's nose. which is why i love her.

Edward Evans

lmao

copper.hat

i'm afraid i may have damaged a few noses & teeth

leslie townes

she just went right into it with the elbow.

copper.hat

not in competition though

leslie townes

which is how to do it, if anyone wants instructions.

i've mashed a few teeth out

copper.hat

05:52

anyone watch mcgregor crap out recently?

leslie townes

elbow is the best way to do it, almost anything is going to hit hard

copper.hat

cringeworthy

leslie townes

that was sad to see.

copper.hat

your elbow is supposed to be super hard (prob urban legend stuff)

Edward Evans

I've never really followed any sports except cycling, didn't even watch the tour this year though

copper.hat

05:53

i was sort of a fan until he started rattling on about families

i like an eclectic mix.

i like watching friends & relatives play

leslie townes

the elbow just goes right in there. the best way is to fake a punch and then spin around and do the elbow. you hear it when it works.

copper.hat

nice!

Edward Evans

A bat hurts too apparently

leslie townes

i probably do not align with conor politically, there seems to be some weird stuff there.

copper.hat

someone came at me with a stick once (in west berkeley) but i blocked it and the stick snapped

the chap was a bit shocked when i started towards him and he ran away

leslie townes

05:55

the people who do that kind of stuff never have a second plan. that was what awakened me to the world of self defense.

copper.hat

i could have tackled him but i had the dinner

besides, it would not have ended well

leslie townes

they have the one thing they think is going to work and when it doesn't work, you've got 5 seconds and look out dude because it's my time.

Koro

Sorry for disturbing the nice discussion. I have a doubt: suppose that $\alpha$ is irrational then I know that there exist positive integer $q_n$ and an integer $p_n$ such that $|\alpha-\frac{p_n}{q_n}|\lt \frac 1{q_n^2}$. From here what can I say about $\lim q_n$ as $n\to \infty$?

copper.hat

what?? maths?

leslie townes

koro thank you for interupting the discussion of how to hit people.

copper.hat

05:56

you mean other than going to infinity?

Koro

will $q_n\to \infty$?

copper.hat

you can find such $q_n$

Koro

I think that will be true if $\frac{p_n}{q_n}\to \alpha$.

leslie townes

i assume you've done the wikipedia stuff

Koro

it's an exercise problem in a book

Edward Evans

05:58

oh

copper.hat

it must be true if the rational quotient converges !!!!!

Edward Evans

I was seconds away from sending a theorem

Koro

the first part was to show existence of ${p_n}$, ${q_n}$ such that $|\alpha-\frac{p_n}{q_n}|\lt \frac 1{n q_n}$

copper.hat

i find it a very peculiar result for reasons i cannot elaborate

$n q_n$???

Edward Evans

You're proving the theorem I was going to send I guess

Koro

05:59

@copper.hat yes copper, that was the first part of the question.

copper.hat

i see where its going

Later

@copper.hat Hello. Can I ask you for a favor?

copper.hat

you can ask...

Koro

from here, that $q_n^2$ on RHS can be also be deduced

Later

Can you tell me how many reopen votes have been cast for this post?

Koro

06:00

but my question is what can we say about $\lim q_n$?

copper.hat

@Later i am not a mod

it looks like no reopen votes from where i am standing

and 1 delete vote

Later

@copper.hat Thanks. Can I know the name of the voter?

leslie townes

similarly i am not a mod i just talk an enormous amount of b---s--- on chat and sometimes answer questions.

copper.hat

@Later i have no idea? how could i tell?

leslie townes

i just talk an ass load of crap.

copper.hat

06:02

sometimes i like ass

Later

@copper.hat You cannot see the delete voter?

copper.hat

@Later you mean the deleve voter? no, it just shows a count?

Later

Yes

copper.hat

why would i see it and you not?

Later

Because I am a low rep user.

copper.hat

06:04

i have no special privileges other than being allowed ro answer convex psqs.

i don't think anyone except for a mod can see such things

leslie townes

my wife and best friend are understanding and realize i am an unhealed person but everyone else has some kind of reason about my posts and thinks, let's keep blocking it, and frankly i'm OK with that.

copper.hat

and i am far from a mod

Later

Ok. Thanks.

Koro

copper, any suggestions that will lead to answer of my question?

leslie townes

copper and i are both in jail.

you're talking to convicts.

copper.hat

06:05

i have tried blocking myself because if find myself disturbing

@Koro sorry, i missed the question?

Koro

lim $q_n$

The book says: since $\alpha$ is irrational, we have $\lim q_n\to \infty$

I didn't understand this statement

copper.hat

if the quotient converges to $\alpha$ then you Must have $q_n \to \infty$,

Koro

10 mins ago, by Koro

Sorry for disturbing the nice discussion. I have a doubt: suppose that $\alpha$ is irrational then I know that there exist positive integer $q_n$ and an integer $p_n$ such that $|\alpha-\frac{p_n}{q_n}|\lt \frac 1{q_n^2}$. From here what can I say about $\lim q_n$ as $n\to \infty$?

copper.hat

well, if $q_n$ is bounded above then you can only get so close to $\alpha$.

Koro

@copper.hat I know that. But from the inequality, is it obvious that the quotient ($\frac{p_n}{q_n}$) converges to $\alpha$?

copper.hat

06:08

it is obvious from the $n q_n$ one, since $q_n \ge 1$.

Koro

Ahh yes!

copper.hat

it is usually an earlier step in a proof

Koro

thanks a lot copper, I understood. I'll take ahead things from here.

copper.hat

excellent!

the there is the bare chested tonga lad who many were drooling over

geocalc33

09:45

13

A: Analytic continuation of $\Phi(s)=\sum_{n \ge 1} e^{-n^s}$

This is currently a partial answer, refining the idea given by @reuns. The series $\Phi(s)=\sum_{n=1}^\infty\ e^{-n^s}$ converges iff $s>0$ is real. Using the Cahen–Mellin integral $$e^{-x}=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(z)x^{-z}\,dz\qquad(x,c>0)$$ with $x=n^s$ and $c>1/s$, w...

The remaining question is whether we can extend this region further.

^I don't understand this. Where else is it possible to extend to?

russian bot

10:47

i just cant stop rolling my r's

cookies

11:01

Hi, could anyone advise whether there is a standardised way to denote two versions of same variable. I am currently using $A_0$ for the initial revision and $A_1$ for the next revision but I wasn't sure if this is right? Many thanks!

satan 29

Hi, I wanted to solve the integral: $$\int_{0}^{\infty} \dfrac{cos(tx)}{x^2+4}dx$$

using feynman's trick

If I denote this by $f(t)$, then $f"(t)$ will be:

$$\int_{0}^{\infty} \dfrac{-x^2cos(tx)}{x^2+4}dx$$

writing $x^2$ as $x^2+4-4$, we can further simplify this into

$$f''(t)=4f(t) - \int_{0}^{\infty} cos(tx) dx$$

But isnt this problematic? the integral obviosuly diverges...

Is there something I did which renders feynman's trick wrong or unusable?

Adil Mohammed

11:54

Oh i thought my doubt was be cleared but its not so, here it is once again

ComFreek

the limit is 0

Adil Mohammed

oh yes e^(-infinity) from the graph, oh yes

Also want to confirm e^-(2/5) is not same as e^(5/2) right?

ComFreek

@AdilMohammed Correct

e^(-5/2) = 1/e^(5/2)

robjohn

12:32

@satan29 differentiating under the integral (Feynman's Trick) does not always work in all situations.

@cookies without any more context, I would say that that is correct.

Adil Mohammed

@ComFreek thank you

robjohn

@satan29 Try $\int_0^\infty\frac{\cos(tx)}{x^2+4}\,e^{-sx}\,\mathrm{d}x$ using Feynman's Trick, and then let $s\to0$. You can get $\int_0^\infty\cos(tx)\,e^{-sx}\,\mathrm{d}x$ by integrating by parts twice, I believe.

shintuku

12:50

"Fix $x \in I$. Then there is a number $M$ depending on $x$ such that $f(x) = P_n(x) + M(c-x)^{n+1}$, where $P_n(x)$ is the nth expansion of the Taylor polynomial at some number $c \in I$."

what's guaranteeing that $M(c-x)^{n+1}$ is a number here for all $x \in I$?

Adil Mohammed

$\lim _{h\to 0}\left(h^{\frac{-2}{3}}\right)$

h--->0^(-2/3) how is it infinity??

or better question maybe, why is 0^(fraction that is<1) infinty?

shintuku

@AdilMohammed notice the negative in the exponent makes you switch numerator and denominator

robjohn

Adil Mohammed

(Assume everything above as delete pls, i cant delete and it and I don't want people to think I forgot exponential formulas)

shintuku

i always forget exponential formulas and im proud of it

robjohn

13:02

@AdilMohammed small numbers to negative powers get big

cookies

@robjohn - no worries thank you! :)

robjohn

@shintuku when would it not be a number?

Adil Mohammed

@robjohn (i am squeezing my brain currently, i remember my teacher telling you wont understand anything in this chapter unless you revise limit. Guess I'll have to take my last years book eeeee)

robjohn

13:18

@AdilMohammed For example, $\lim\limits_{n\to\infty}\left(\frac1n\right)^{-1}=\lim\limits_{n\to\infty}n=\infty$

shintuku

13:44

@robjohn oh, I realized that, since there is always a number that expresses the difference between a function and it's taylor polynomial, that expression is fine

thanks for the hint

Srijan M.T

13:55

In differentiation of tan^4(x)

I used chain rule

Let t = x.

dy/dt = 4(tan^3) * t

dt/dx=1

So , answer is 4(tan^3) * x

But online , answer is different

Where am I going wrong

Wolgwang

14:24

@SrijanM.T $$y=\tan^4(t)\\\frac{\mathrm d y}{\mathrm d t}=4[\tan(t)]^3\times \frac{\mathrm d}{\mathrm d t}[\tan (t)]$$

Srijan M.T

@Wolgwang Ok thanks

@Wolgwang If integration of x^2 = x^3/3 +C. Then ,differentiation of x^3/3 +C is not coming x^2.

How come this is not true ?

Akiva Weinberger

Hey

Wanna watch a two-hour unedited Zoom conversation about knot theory?

2

Unedited knot theory discussion with Shayna

►

In which I teach a mathematician friend about knots

Wolgwang

@SrijanM.T It is $x^2$...

Srijan M.T

@Wolgwang

I get is 2x^2 / 3 by using division rule.

Can you tell where is the mistake or just in brief what to do @Wolgwang

Wolgwang

14:46

@SrijanM.T It is $(\frac{x}{3})^3+c$ not $\frac{x^3}{3}+c$

Srijan M.T

Ohkk @Wolgwang

Thanks a lot.

SAJW

15:08

mmh, $1^x$ can be 2

interesting

mmh, is it true: $1^x=2 \Leftrightarrow 1=2^{\frac{1}{x}}$, sure if x goes to infinity the right side is true, but feels wrong

Bob

I have done the same Calculus problem three times. I still get the wrong error. I think it is an algebra error. What is the best way for me to find the error? Any ideas?

satan 29

15:26

@robjohn but why

or more appropriately. what was in that particular integral that made it fail

@robjohn well the last integral is just the laplace transform of cos(tx) so its a known integral...

copper.hat

16:41

@satan29 you cannot apply the Leibniz integral rule blindly. the integrand in $f'$ is not integrable.

@SAJW for $x \neq 0$ we have $1^x = 1$, so the equivalence is meaningless.

@Bob you must be making a mistake. if you do not want to consider other approaches then you have little recourse other than to revisit your computations.

satan 29

16:54

@copper.hat how did you deduce that?

SAJW

@copper.hat I don't know enough about complex numbers, so is it bogus or valid? youtube.com/watch?v=9wJ9YBwHXGI&ab_channel=blackpenredpen

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$Mathematics$ Mathematics