Mathematics - 2012-10-02 (page 1 of 3)

user19161

01:54

I am going to bed. Good night @peter @jayesh!

robjohn

02:41

@JasperLoy good night

only missed by 50 minutes

Yee-haw! I have 900 answers :-)

Jayesh Badwaik

05:37

@JasperLoy night. Late by 4 hours. :P

@PeterTamaroff How did you get this?

9 hours ago, by Peter Tamaroff

$$\frac{1}{{1 + {x^2}}} = \frac{1}{{1 + {x^{4n + 2}}}}\sum\limits_{k = 0}^{2n} {{{\left( { - 1} \right)}^k}{x^{2k}}} $$

@robjohn nice. :-)

robjohn

@JayeshBadwaik Thanks!

Using the sum of a geometric series, we get
$$
\sum_{k=0}^{2n}(-1)^kx^{2k}=\frac{1+x^{4n+2}}{1+x^2}
$$

Jayesh Badwaik

05:56

@robjohn Ohh. Alright.

@OldJohn India alone adds a population equal to the whole of australia every year, the statistics were correct 10 years ago, the last time the census was taken in India,now I am guessing it will be even more. Actually last time the census was taken was in 2011, but the data has not been completely analyzed from that one I would guess.

Old John

07:29

@JayeshBadwaik As often, my knowledge is out of date :)

robjohn

07:42

@OldJohn we are simply out of date :-)

Old John

@robjohn But not past our sell-by-date - our experience usually compensates :)

robjohn

@OldJohn sure. We are the Elders

Old John

@robjohn :)

Ilya

08:41

@robjohn: are you around?

robjohn

@Ilya Hey there! haven't seen you in a while :-)

Jayesh Badwaik

10:12

How can browse questions on which a bounty was offered but has probably now expired or was rewarded? Is there any way to do that?

user19161

12:18

I think this question will hit the multicollider soon. math.stackexchange.com/questions/205953/…

Old John

@JasperLoy pardon my ignorance, but what is the multicollider?

user19161

@OldJohn Ah, it is a term used on SE to mean the list of hottest questions that appear on stackexchange.com. You can see them in your "global inbox" on the top left when you are logged in to any site as well.

Old John

@JasperLoy Ah - thanks

user19161

@OldJohn The hottest questions from ELU are usually those that involve sex.

Old John

@JasperLoy what a surprise - and those from MSE don't :)

user19161

12:34

I gave another variation of the answer.

user19161

12:57

Well, on second thought, it is too similar so I deleted it. =)

user19161

There are only three people in this room now, amazing!

Jayesh Badwaik

13:22

@JonasTeuwen Thanks a lot for the advice to use mendeley.

@JasperLoy now there will be only two, leaving for dinner, bye.

bbl

Dan the Man

13:42

Hey guys

I have to solve this with the quadratic formula.

$4x^2-2x-14=0$

I got $x=\frac{2+2\sqrt{57}}{8}$

As you can see, It is irrational.

My math book says, "Express the roots exactly using fractions and radicals. In the next blank, express the roots as decimals, rounded to the hundredths' place."

How can I simplify my answer so that it is a simpler fraction? What do they mean by "Express the roots exactly using fractions and radicals"?

Wannabe Mini Mathematician

14:00

Good evening.

Hello @JayeshBadwaik. We haven't met for...

Dan the Man

Good morning.

Jayesh Badwaik

@WannabeMiniMathematician Hello.

long time no see. Also, change your screen name!!

Wannabe Mini Mathematician

@Jayesh: Wait till October 25th. :(

Jayesh Badwaik

@WannabeMiniMathematician ohh.

Jonas Teuwen

@JayeshBadwaik Now we should write a nice plugin for emacs.

So that the AucTeX bindings go look in the SQLite DB (better than the API for this), and if found exports the bibtex and stores it locally.

Wannabe Mini Mathematician

14:16

There exist two types of people:
1) Those who are in this category.

2) Those who are not.

Helmut

Not if your category is a fuzzy category

Dan the Man

14:37

Hello again...

$x^2-c=0\\
x\stackrel{?}{=}\sqrt{c}
$

Jayesh Badwaik

@JonasTeuwen For now, what I have got is good enough for me, in the sense that it is not hindering my work as of now. If I want to go further, I have some other bigger plans. For example, implementing a tagged file system.

Yes, we do need an auto-inserter though, it should be fairly simple, I don't know too much Emacslisp however.

@DantheMan duh!

Dan the Man

Why in the world did my mom mark this one wrong???

Helmut

$\pm$

Dan the Man

14:53

@Helmut Isn't that implicitly there?

Helmut

No?

Jayesh Badwaik

@DantheMan No, $\sqrt{c}$ indicates the positive square-root of c.

Dan the Man

@JayeshBadwaik Oh ok. thanks.

Jayesh Badwaik

@DantheMan Also something from long ago.

Sep 11 at 13:07, by Jayesh Badwaik

NEVER FORGET!!

Helmut

I should have gotten that thank you. What a gyp.

Dan the Man

14:55

@JayeshBadwaik Hahaha yes... How did you find that so fast?

Jayesh Badwaik

Google search.

site:chat.stackexchange.com "NEVER FORGET"

Dan the Man

Wow. I wouldn't have thought of that.

Jayesh Badwaik

Its no biggie, you would learn with age.

Dan the Man

$-12=4\sqrt{x^2-7}$ now what?

Square both sides?

Peter Tamaroff

15:36

@DantheMan How can a sqrt be negative??

(well, it can, but....)

Dan the Man

@PeterTamaroff Huh?

Peter Tamaroff

$-3=\sqrt{x^2-7}$

@DantheMan We usually consider positive sqrts, but we're OK here.,

So you want $x^2-7=9$

Dan the Man

Yes.

bingo

I already got it, but thanks

Wannabe Mini Mathematician

And it gets easier when you have $x^2 - 16 = 0$

Voila! $(x + 4)(x - 4) = 0$

Dan the Man

Yeah!

Jonas Teuwen

15:47

@JayeshBadwaik Tagged file system? Did you read my ramblings about that?

I have it!

I wonder how you would want to do that. I'd use FUSE.

Anyway, the emacs thingie can be quite cool.

Awesome actually.

Jayesh Badwaik

16:26

@JonasTeuwen Yup, emacs plugin would be awesome. I read your rambling about the tags, I would implement in fuse too (or can be implemented even lower). My design is quiet similar to yours, however, I have some ideas which may make it possible to avoid the sql and hence, allow for a completely independent code. SQL implementation is too inviting for problems. Let's see.

@WannabeMiniMathematician 9 overs 40 runs? What do you think? Can we defend it?

Jonas Teuwen

@JayeshBadwaik I agree with the SQL. There is little relation needed.

I would not implement it any lower as you would still like to have the files in a "normal file system" which can handle the type of files you want really well. All you need to implement is a kind of mapper.

Old John

Hi all

Jonas Teuwen

I have decided to ask a question and display my ignorance even more.

Hi.

And preferable also expand it.

Jayesh Badwaik

@JonasTeuwen Yes, me thinks too. But the design I have been working out can be implemented at a lower level too. So, I may not implement it, but I just think it can be.

Peter Tamaroff

"My question is about the tensor product of Banach spaces but actually I do not understand those either, so let me rephrase it as a Hilbert space question." XD

Jonas Teuwen

16:36

I don't see the need to do any kernel interaction which the FUSE module doesn't do.

Jayesh Badwaik

@JonasTeuwen Yup. I agree.

Jonas Teuwen

@JayeshBadwaik A NoSQL thing might be better as what you would like to do with tagging is like a "real life mapping" of your files.

Without relational stuff. You just put it in a file.

And that can be: bookshelf, but also the box on the bookshelf.

You don't put a note in some box where you go first that says "look there!" and then there again...

I would implement those as some extended file properties.

But then you might need lower level, is that what you mean?

Jayesh Badwaik

@JonasTeuwen Yes, that is what I meant.

Jonas Teuwen

Those properties would be like a JSON thingie.

Yes, but you could perhaps do that two-fold: the normal file system and then add the extended file properties (already implemented).

And then a userland file system that just makes tags possible.

Which would read the extended file properties from the underlying file system.

No shit with SQL :-).

@PeterTamaroff Yes.

Jayesh Badwaik

Yup.

Jonas Teuwen

16:40

That was the idea as well?

While we are at it, we can use python bindings to FUSE.

And do the whole thing in python!

Peter Tamaroff

@JonasTeuwen this is Dog.

Wannabe Mini Mathematician

Goodbye India. Try out in 2013.

Jayesh Badwaik

@WannabeMiniMathematician du Plessis goes but damage done.

Jonas Teuwen

@PeterTamaroff What?

Wannabe Mini Mathematician

@JayeshBadwaik: :(

Jayesh Badwaik

16:43

@JonasTeuwen Yup, exactly what I was thinking. I have done some code in C++, but python might be better.

Jonas Teuwen

@OldJohn Can you help me in ignorance? 8-).

No need for C++, I would say.

Might be harder, Python has dictionaries.

Old John

@JonasTeuwen I think my ignorance is much greater than yours!!

Jonas Teuwen

@OldJohn Oh?

@JayeshBadwaik And a python dictionary is basically just a thing with "here is your stuff!"

Old John

But I am always ready to listen :)

Jonas Teuwen

But... I don't know how good the extended file system properties are, you might have to store some extra information elsewhere. That would say "here is the other stuff".

@OldJohn Oh, I have linked it a bit higher :-).

Jayesh Badwaik

16:46

@JonasTeuwen Hmm. Yup. So, for the precise reason, I have a little different design that does all of that and some more. It is basically, "here is your stuff, this is the stuff related to your stuff and this is the stuff your stuff is related to (which is a different thing)"

Wannabe Mini Mathematician

I'm afraid, but this one, again, is a bad chemical joke.

@JayeshBadwaik: 14 more...

Jayesh Badwaik

@WannabeMiniMathematician Its not caustic, so its okay.

Jonas Teuwen

@JayeshBadwaik I like the idea. Can I help?

Wannabe Mini Mathematician

@JayeshBadwaik: LOL!

Still we have hopes to count on!

Old John

@JonasTeuwen I'm still trying to understand your new question :)

Jayesh Badwaik

16:50

@JonasTeuwen Of course. Actually, in the end you might end up contributing more to it, so think of it as an equal colloboration. I am writing out my ideas, will upload them as soon as I am done writing.

Jonas Teuwen

@OldJohn It is no good? 8-(.

@JayeshBadwaik On github! 8-).

Just the README.

Jayesh Badwaik

@JonasTeuwen Yup!

Old John

@JonasTeuwen I am sure it is good - but I am struggling

Jonas Teuwen

Hmm... I thought it was quite simple. What is not to be understood?

Old John

@JonasTeuwen tensor products!

Jonas Teuwen

16:52

@OldJohn That is just a word I use to make me look smart. I don't know what those are!

Old John

@JonasTeuwen me neither :(

Wannabe Mini Mathematician

@JayeshBadwaik: What if India wins?

Jayesh Badwaik

@WannabeMiniMathematician doesn't matter. We have to win by 31 runs. We will go out on NRR.

Wannabe Mini Mathematician

omg

Jayesh Badwaik

another wicket goes.

Jonas Teuwen

16:53

@OldJohn You are not alone...

Jayesh Badwaik

@WannabeMiniMathematician but the situation is hopeless now.

Wannabe Mini Mathematician

@Jayesh: Yeah, I was wondering if India could really go within 121.

oh no

no ball

=(

Jayesh Badwaik

I am not watching anymore.

Old John

@JonasTeuwen :)

Peter Tamaroff

@JayeshBadwaik What are you talking about?

Sports?

Old John

16:56

Cricket?

Jayesh Badwaik

@PeterTamaroff cricket. T20 world cup. India are going out (unless there is a miracle and rain comes in or something).

Jonas Teuwen

@OldJohn Indians. What else? 8-).

Jayesh Badwaik

@JonasTeuwen :P :P

Old John

@JonasTeuwen and British are also crazy about the game ...

Jonas Teuwen

I know... 8-).

Jayesh Badwaik

17:04

@JonasTeuwen What about Netherlands? apart from eredivisie?

Jonas Teuwen

Nothing else.

Jayesh Badwaik

I mean apart from football?

Jonas Teuwen

Rowing perhaps.

And ice skating.

Son of a holy cow, nobody even commented on my question 8-(. Is it that bad?

Jayesh Badwaik

I like dutch league football, its entertaining to watch. Quiet more free flowing than british.

Jonas Teuwen

Mmm, I have never watched it.

Jayesh Badwaik

17:06

@JonasTeuwen ???

user19161

@JayeshBadwaik There can be miracles when you believe.

Jayesh Badwaik

@JasperLoy Unfortunately, in a game, both sides can believe.

Old John

Going out to eat - back in a while

Jonas Teuwen

@OldJohn Enjoy.

Charlie

17:32

Hello!

@JayeshBadwaik

@JasperLoy

@JonasTeuwen

@robjohn

Jonas Teuwen

8-).

Charlie

Hey,jonas!

Jayesh Badwaik

@Charlie boa tarde

(if that is right)

Charlie

Jonas, you said that you drink in the chess club,right?Have you ever played drunk?

@JayeshBadwaik it it ,it is!!!!!!!!!!\o/

@JayeshBadwaik Shubh Ratri (is that right?)

Jonas Teuwen

@Charlie Yes, I have.

Charlie

17:40

@JonasTeuwen hehehe

Jayesh Badwaik

@Charlie Yup.

Charlie

@JayeshBadwaik YAY!!oh my!

Jayesh Badwaik

Charlie

@Argon hi!

@JayeshBadwaik it's a nice language!

Jayesh Badwaik

@Charlie However, shubh ratri is used for good night. Shubh is Good. Ratri is night. We generally say good evening though which is shubh sandhya. Sandhya means evening. Sandhya is also a girl's name though.

Argon

17:42

@Charlie Hey

Jayesh Badwaik

@Charlie It is. Did you look at the vodka chess?

Charlie

@JayeshBadwaik yes i did.haha

When you are close to checkmate...you are probably close to a coma

Jayesh Badwaik

whenever a piece is killed, the opponent player (the one who killed it) has to drink the contents of that piece.

Charlie

@JayeshBadwaik jeez!

Jayesh Badwaik

@Charlie ensures that the game is balanced throughout. :P

Argon

17:46

It seems to me that the subject of drinking arises very often around here

Charlie

well, i don't drink.i like my liver

@JayeshBadwaik hehe

@JayeshBadwaik i remered you yesterday

Jayesh Badwaik

@Charlie how?

Charlie

@JayeshBadwaik remembering :P

i understood something that i wasn't...

Jayesh Badwaik

@Charlie Hmm. :-)

Argon

Is it okay to write $\left(\frac{1}{2}\right)!$ or is $\Gamma\left(\frac{3}{2}\right)$ always preffered?

Charlie

17:51

Hey @Jas!

user19161

@JayeshBadwaik I got a glass chess set once in a lucky draw, but I gave it away because I don't play chess anymore and even if I did, I would not use it.

user19161

@Charlie Hello M.

Charlie

@JasperLoy did you read my mail?

user19161

@Charlie A little bit of drinking is fine, for example, a glass of wine a day.

Jayesh Badwaik

@JasperLoy Hmm. All chess that I play is against a computer currently.

user19161

17:52

@Charlie Not yet.

Charlie

@JasperLoy i rather not

Jayesh Badwaik

@JasperLoy I wanted to say the same. A little bit of drinking is actually good for your heart (especially if you are older) and does not affect liver.

user19161

@Charlie I used to be averse to alcohol too. After my friend gave me a bottle of wine as a present, I drank a little bit each day, and then I started to like it. But I usually don't drink.

user19161

@JayeshBadwaik Too much water can kill a person too! All things in moderation.

Charlie

hmm

user19161

17:55

Hey guys, my new phone is so complicated that I have not figured out how to use it.

Jayesh Badwaik

@JasperLoy it takes you three to four days of full study to work stuff out.

Charlie

@JasperLoy hehe you get used to it...

user19161

@JayeshBadwaik I am getting scared. I might even switch back to the old one now. But I have not read the manual.

Jayesh Badwaik

@JasperLoy The problem here is I have seen too many alcoholic cases related domestic issues. Whenever there is talk of alcohol being good or bad, they hover in back of my mind. I don't have anything specific against alcohol, I even like it somewhat, but those examples are always a warning in my head.

user19161

@Charlie Hmm, I have a few thoughts which I would keep to myself now, but maybe one day I will tell you what I am thinking...

Charlie

17:57

@JasperLoy is it good or bad?

user19161

@JayeshBadwaik It is not alcohol that makes a person evil. It is an evil person that makes alcohol evil.

Charlie

hey , would you like to listen some good brazilian song?

Jayesh Badwaik

@JasperLoy of course.

@Charlie yup.

Charlie

here lyrics in description

user19161

There are three pink avatars side by side, amazing.

Jayesh Badwaik

17:58

@JasperLoy alcohol is like high speed, there is nothing inherently wrong with it. It just might make things difficult to control.

Argon

@Charlie Where did you learn English?

Charlie

@Argon school...why?

and a lot aby myself

Argon

@Charlie Its good

Impressive

Charlie

@Argon Thanks!

user19161

@Charlie I don't know.

Charlie

18:00

@JasperLoy how you don't know?

user19161

@Charlie They are neither good nor bad to me, but I don't know how you will see them.

Charlie

@JasperLoy hmm

this is the best brazilian band ever!!!!

user19161

Besame, besame mucho...

Charlie

bolero

Legião urbana is the best band that brazil ever seen

good lyrics,most of them about dictatorship

Jayesh Badwaik

@Charlie but you have a democracy now, don't you?

Charlie

18:04

@JayeshBadwaik hmm...i still think we live in a dictatorship..but people don't realize it...

Jayesh Badwaik

@Charlie okay. that way, India is still a monarchy.

Charlie

this is my favorite!

user19161

This world will be full of shit until I take over!

Charlie

@JasperLoy do you like that song?

user19161

@Charlie I heard Andrea Bocelli sing it.

Charlie

18:06

@JasperLoy it's nice

this one is very beautiful!

I love this singer!

David K.

Can someone tell me how to "cross-through" text in a MSE post?

Jayesh Badwaik

~~doesnot~~

@DavidK. ""~~doesnot~~""

put three dashes in the front and at the end

Charlie

@Argon wassup?

Jayesh Badwaik

"""~~doesnot~~"""

David K.

@JayeshBadwaik hrmmm.... the text isn't showing up as struck-through in the preview...

Charlie

18:19

@jas

David K.

@JayeshBadwaik Nevermind. I got it. It's "<del>" and "<fwdslash del>"

robjohn

@Ilya Next time you're around, we should talk ;-)

Jayesh Badwaik

@DavidK. okay. good. Basically this site uses markdown formatting system, so in case you want to know more about it, you can just google, its very popular. I am not too familiar with it.

@Charlie

Charlie

@JayeshBadwaik yes?

David K.

@JayeshBadwaik Yes. I just realized that. Sorry, should have googled that first. I didn't realize that the formatting system had a name.

Jayesh Badwaik

18:25

@DavidK. Its okay. You don't need to apologize, I let you know, because I myself cannot use it, and as you saw, supplied you the wrong answer (which works in the chat), I don't know why.

Charlie

did you guys know that in south of my country there's a city called "Holambra"?a mix of holland, america and brasil.the city of flowers.

Jayesh Badwaik

@Charlie no i did not know that.

Charlie

@JayeshBadwaik the south is really influenced by europe

there are many germans too

Jayesh Badwaik

@Charlie okay. May be because the climates match more?

Charlie

@JayeshBadwaik i really don't know...

there's a tiny city .it's more ucranian than brazilian.people learn old ucreanian in schools

Jayesh Badwaik

18:33

@Charlie ohh.

Charlie

@JayeshBadwaik my state is more influenced by itailians

Jayesh Badwaik

@Charlie okay.

we had british run all over us, so except for goa and some other states, everything is british.

Charlie

Pedro!

Peter Tamaroff

@Charlie Sup?

Charlie

@PeterTamaroff i'm good.ya?

Peter Tamaroff

18:35

@Charlie Can't complain

Charlie

@PeterTamaroff good

Peter Tamaroff

@Charlie Weren't you having some exam soon?

Charlie

@JayeshBadwaik can i make a silly silly question?

@PeterTamaroff i had it

last friday

Nimza

Good evening!

Charlie

@Nimza hi !!

Argon

18:37

@PeterTamaroff Where did you learn English so well?

Nimza

@Charlie hello :)

Peter Tamaroff

@JayeshBadwaik I was suspecting all monotone functions on closed intervals are bounded. Seems it is not too hard to prove.

Jayesh Badwaik

@Charlie yes you can.

Nimza

Who does like the game "devinette"? :)

Jayesh Badwaik

@PeterTamaroff monotone continuous?

$tan(x)$ ?

Peter Tamaroff

18:38

@JayeshBadwaik Just monotone.

Charlie

@JayeshBadwaik have you ever watched a movie called:"Saawariya"?

Peter Tamaroff

@JayeshBadwaik CLOSED INTERVAL!

Jayesh Badwaik

@PeterTamaroff Yeah, domain is $[0,\pi/2]$

Nimza

Peter Tamaroff

@JayeshBadwaik That is not defined on $\pi/2$, dude.

Jayesh Badwaik

18:39

@Charlie I have loved that movie. It is based on the book "White nights". If you loved it too, you are the only other person I know.

Peter Tamaroff

@JayeshBadwaik At most $[0,\pi/2)$

@Argon I studied English for 12 years, kinda like "high school" things...

Nimza

Do you know what is it?)

Peter Tamaroff

then GCSE exams.

Charlie

@JayeshBadwaik it's cute.I liked more this part

Jayesh Badwaik

@PeterTamaroff Hmm. Okay, but I still think, discontinuity should create a problem.

Charlie

18:40

@JayeshBadwaik didn't know ita was a book.i thought a little sad :(

Peter Tamaroff

@JayeshBadwaik I tried to come up with examples, but it seems I need more imagination, or LSD...

Charlie

just a sec.BRB

Jayesh Badwaik

@Charlie this is my favourite track in the movie. The song-dance is kind of cliche, but the music is good,

@PeterTamaroff Hmm, let me see if I can come up with something. If not, then let me see if I can come up with a proof.

Charlie

@JayeshBadwaik yeah.it's nice.

BRB

Peter Tamaroff

@Argon Sorry, not 12, 8

Argon

18:43

@PeterTamaroff Still a lot, eh?

Peter Tamaroff

@Argon Guess so.

@JayeshBadwaik Spivak sketches one.

Jayesh Badwaik

@PeterTamaroff okay, where? I mean page number?

Peter Tamaroff

@JayeshBadwaik PG?

Jayesh Badwaik

I meant reference.

Peter Tamaroff

@JayeshBadwaik Oh. In my edition, it is exercise 21 from chapter 13

Jayesh Badwaik

18:48

It is kind of obvious now.

$f(a)$ is finite.

$f(b)$ is finite

in the interval $[a,b]$

And since the function is monotone.

$f(x) \in [f(a),f(b)]$ ( WLOG $f(a) < f(b)$)

Hence, proved.

Peter Tamaroff

@JayeshBadwaik ?

robjohn

@Nimza donut-filled donuts!

Jayesh Badwaik

@PeterTamaroff We wanted to prove that $f(x)$ is bounded on $[a,b]$ right?

Nimza

@robjohn yeah) 2-adic solenoid

Peter Tamaroff

@JayeshBadwaik Duh! Not bounded, integrable =)

Jayesh Badwaik

18:51

@PeterTamaroff duh, that is why I am thinking, is there a hole? :P :P I thought it was some trick question or something.

Peter Tamaroff

@JayeshBadwaik I think I see now.

Jayesh Badwaik

@PeterTamaroff okay. I think it is integrable, but getting the bits right might be a little trickier.

Basically it is telescopic thing.

Peter Tamaroff

@JayeshBadwaik Kinda. Let me show you

Jayesh Badwaik

Since the function is monotone, you have the max and min at the end of the interval

and hence, they cancel out.

when you subtract the U from the L

Peter Tamaroff

Let $P=\{t_0,t_1,\dots,t_n\}$ be a partition of $[a,b]$. Let $M_i$ and $m_i$ be the infimum and supremum. Then you get the following:

Argon

18:57

$א_0$

Peter Tamaroff

$$\eqalign{
& \sum\limits_{i = 1}^n {f\left( a \right)\left( {{t_i} - {t_{i - 1}}} \right)} \leqslant \sum\limits_{i = 1}^n {{m_i}\left( {{t_i} - {t_{i - 1}}} \right)} = L\left( {f,P} \right) \cr
& U\left( {f,P} \right) = \sum\limits_{i = 1}^n {{M_i}\left( {{t_i} - {t_{i - 1}}} \right)} \leqslant \sum\limits_{i = 1}^n {f\left( b \right)\left( {{t_i} - {t_{i - 1}}} \right)} \cr} $$

RIght? @JayeshBadwaik

Jayesh Badwaik

@PeterTamaroff Correct.

Transcript for

$Mathematics$ Mathematics