War Room - 2017-11-28

Anyway, last month, I felt really good after I finished most of the CTPC inequalities chapter. Towards the end I felt like I was learning to think a little like an analysis person.

So what are you doing now? Studying really hard?

Balarka Sen

I have been doing a chunk of number theory and combinatorics

so we have our roles reversed

@SohamChowdhury nah

i can't study hard

Soham Chowdhury

indeed, indeed

Balarka Sen

im biologically retarded

Soham Chowdhury

I am going to do a bit of "number theory" now

Balarka Sen

What's in CTPC inequalities?

Soham Chowdhury

3:08 PM

oh, not much. C-S, weighted AM-GM, a few inequalities that you can derive from them, etc.

mostly it's just clever algebra

Balarka Sen

ah-hah

Soham Chowdhury

but CMI is less inequality-heavy than ISI, say

Balarka Sen

Right

Soham Chowdhury

it's really all the "logic problems" that get you
prove this algorithm terminates, yada yada

Balarka Sen

I actually learnt to inequality-manipulate in my statistics courses in school

I'll have a look at CTPC's ineq chapter rn

well flick

libgen doesnt have it

Soham Chowdhury

3:10 PM

room topic changed to War Room: (no tags)

Feeds

Soham Chowdhury has added Balarka Sen to the list of this room's owners.

Soham Chowdhury

room mode changed to Gallery: anyone may enter, but only approved users can talk

Balarka Sen

danks

Soham Chowdhury

why did you think libgen would

Balarka Sen

iunno

Soham Chowdhury

3:12 PM

anyway, I will be ordering a copy of Engels again

I lost mine a few years back, it is Good

Balarka Sen

I have le Engels

it's quite nice

if only i read anything out of it

Soham Chowdhury

especially the combi chapter taught me how to think, but i've forgotten lmao

Balarka Sen

i'm shocked!!!1!

Soham Chowdhury

ah, it's good to be back

Balarka Sen

Truly

How is college?

Soham Chowdhury

3:14 PM

Bad

The analysis prof is good, but the others are stupid

Balarka Sen

?

oh

Soham Chowdhury

Analysis guy knows his stuff, but, e.g. algebra teacher calls sets with just an operation on them "groupoids"

Balarka Sen

arent those magmas

Soham Chowdhury

linear algebra teacher says $(A,B) \mapsto \frac12 (AB-BA)$ is nonassociative because "matrix multiplication is noncommutative"

what if I change the $-$ to an $+$? "stop arguing"

@BalarkaSen yip

Balarka Sen

lol

welp

I learnt a lot of linear algebra

feelsgoodman

Soham Chowdhury

3:16 PM

and I had a really bad midterm because linalg teacher said "wtf is a two-sided identity", etc. and everyone marked me badly

Balarka Sen

not sure if it's of any use to CMI schtick though

Soham Chowdhury

what kind of thing?

Balarka Sen

Down to earth linear algebra which I knew only in half-assed format for all these years

Soham Chowdhury

half the tests here are Gaussian elimination

Balarka Sen

Structure theorem of matrices, etc etc

Soham Chowdhury

3:17 PM

oh, yes, that is a good feeling

remember when I learned multicalc

Balarka Sen

@SohamChowdhury Wait, what?

Soham Chowdhury

basically, the teachers here mark papers by comparing them to "known answers" in their heads

you'd think a linear algebra teacher would know that one-sided identities exist

"don't act smart"

I can't not write like that anymore :(

anyway

let's get this ball rolling

I have to leave this shithole

Balarka Sen

Yes. Yes you have to

and i have to not fall into it

this place sucks

Man I'd be glad if I at least get ISI

Soham Chowdhury

Apparently you can, like, go to IMSc all the time

someone (on Quora, where else) mentioned a first-year getting credit for a diffgeo course at IMSc

you should be that guy wink wink

Balarka Sen

I don't really care about going anywhere, I just want to get into a decent enough university where I can continue studying math

ISI is decent enough that I would not have to unlearn the math I know already to survive college

can u screenshot me some exercises from CTPC ineqs

if possible

Soham Chowdhury

3:48 PM

sorry, was away

Balarka Sen

kk

Soham Chowdhury

I can take pictures, yes

Balarka Sen

cool

Soham Chowdhury

will send you some on gchat later.

what are you doing now?

Balarka Sen

nice

@SohamChowdhury vaguely fiddling with some problems about polynomials in Excursion

Soham Chowdhury

3:50 PM

gimme one or two pls, I'm bored but also don't want to get up

Balarka Sen

I also want to do some actual math

@SohamChowdhury Ah OK let's see

Soham Chowdhury

I have done zero actual math for a very long time

just looking at le algebraic geometry book once a month like it's some ritual

:(

Balarka Sen

I'll just send you whatever. Here's one which says "Find all poly's satisfying $p(x+1) = p(x) + 2x + 1$"

I have an idea and it's not hard

yeah ok i got it i think

Ugh the book does it by normie techniques

Soham Chowdhury

now you will realise how badly I suck

maybe differentiating will work?

Balarka Sen

Don't worry about it. I used an idea that is a personal favorite of mine; we can work together so we both improve at the stuff we succ

@Soham Close

Soham Chowdhury

3:57 PM

"succ"

I see you are a man of culture etc

Balarka Sen

You can read that off from my profile description in chat

I have convinced myself that memes are true form of art

thou art meme

Soham Chowdhury

sometimes substituting roots of unity works well

Balarka Sen

hm

Soham Chowdhury

this doesn't look like that though

Balarka Sen

yeah

i'll give you a hint. i think about forward differencing a lot

Soham Chowdhury

4:02 PM

I'm trying something like $\sum_n a_n ((x+1)^n-x^n) = 2x + 1$ rn

anyway, you don't have to get me to solve anything now. i'll remember, and maybe tell you I've got it after two days :)

Balarka Sen

That's cool by me

Soham Chowdhury

throw a few more at me if you like, that way I have something to do the rest of the evening

(btw, I really need to learn geometry, I don't know shit about that. somehow I think that not all Euclidean geometry can be $\mathbf C$-ed away, lol.)

Balarka Sen

Ok, let me find some more

I haven't done anything beyond this problem so far so I'll be random

@SohamChowdhury Oh me too man

I don't know heck about Euclidean geometry. I suck at it

I can do coordinate reasonably well though

Soham Chowdhury

@BalarkaSen at this college I'm having to do a lot of "classical" coordinate geometry

if I stay here for three years I'll turn into a fucking 19th century Italian geometer

@BalarkaSen shoot

Balarka Sen

Doing multivariable calculus taught me a lot of vector geometry, and I'm glad for it

fduck the internet

Hmmm this looks like a weird ass problem

For any positive integer $n$, prove that there exists a polynomial $P(x)$ of degree at least $8n$ such that $$\sum_{k = 1}^{(2n+1)^2} |P(k)| < |P(0)|$$

o lawd how do i

Soham Chowdhury

4:17 PM

this looks like a complex analysis problem, but then that's how I react when I see inequalities of this nature

is this Excursion?

Balarka Sen

hm

yep

The first thing to notice is that I just want to bound $| \sum_k P(k)|$ anyway

Soham Chowdhury

... doesn't that weaken the inequality? after all $$\left|\sum P(k)\right| \le \sum |P(k)|$$

Balarka Sen

and it's easier to work with $|\sum P(k)|$

Soham Chowdhury

yes, but bounding the easier expression doesn't imply bounding the given one ... am I being dense?

Balarka Sen

I don't mean that if I bound the mod-sum by |P(0)| then I bound sum-mod too

no, no, you're right to be confused. I didn't mean that

Soham Chowdhury

4:22 PM

you meant testing hypotheses?

Balarka Sen

ya

Soham Chowdhury

ah, got it

Balarka Sen

I write down trivialities a lot. Sometimes they lead to a good guess

In this case I want to construct a $P$ so that seems like a good approach

Do you think it's a good idea to do this for a specific $n$?

as a test case

This is fucking strange so I have no idea

Soham Chowdhury

$8$ is big

it says Z^+ but you can try 0

Balarka Sen

how about $n = 1$... lol

Soham Chowdhury

4:26 PM

actually, wait, that won't work

(the lhs is then empty, lol)

Balarka Sen

mhm

Soham Chowdhury

"degree at least 8n" wtf

Balarka Sen

so I want a $P$ of degree at least $8$ such that $\sum_{k = 1}^9 |P(k)| < |P(0)|$

@Soham the book makes you be like that sometimes

Soham Chowdhury

I think maybe most of the coefficients will be 0, idk

Balarka Sen

prolly, prolly

wait, $P(0)$ is the constant coefficient of $P$

Soham Chowdhury

4:28 PM

wait

Balarka Sen

do you not think it's kind of weird to be bounded by the constant coeff?

Soham Chowdhury

you can probably triangle-inequality things

$|\sum P(k)| \le \sum |a_i k^i| = \sum |a_i| k^i$

Balarka Sen

What if I let all the coefficients to be the same

Soham Chowdhury

collect all the terms with the same $a_i$, you get a sum of the $k^i$ which you can bound somehow

Balarka Sen

like, all of them are the constant coeff, $P(0)$

right

Soham Chowdhury

4:30 PM

that is my best guess for a solution unless some trick like this ^^ holds

Balarka Sen

So, $P(x) = P(0)(1 + x + \cdots + x^N)$?

i.e., $= P(0) (x^{N+1} - 1)/(x - 1)$

for some $N$

Well $\sum_k (k^{N+1} - 1)/(k-1)$ is going to be fucking huge m8

Maybe I have to divide $P(x)$ by something big ass

Soham Chowdhury

I have to eat

cya

Balarka Sen

bubye

Soham Chowdhury

(oh, btw, the $p(x+1)$ problem gives a kind of telescoping sum that you can use for integers)

Balarka Sen

Yep

If you do it for all integers you also do it for all real numbers

'cuz polynomial is determined by it's values on a finite number of integers

Soham Chowdhury

4:37 PM

yep

Balarka Sen

half the problems in one chapter in Niven-Zuckermann-Montgomery were doable by forward differencing and telescoping

good times

Soham Chowdhury

that inequalities chapter too

Balarka Sen

ahh

Balarka Sen

4:50 PM

Heyyyy, that's pretty good

Hmm, what if I use $p(x) = (x - 1)(x - 2) \cdots (x - 8)$

$|p(0)|$ is just $8!$

And $p(k)$ for $k = 1, \cdots, 8$ is just $0$

but eh

Oh no that almost works, $p(9) = 8!$

I am guessing something of this sort is going to do ze trickery

Well, yeah, that immediately gives you a solution

'cuz you are allowed to have degree greater than $8n$

Transcript for

♦ $Mathematics$ War Room

Transcript for

♦ War Room

♦ $Mathematics$ War Room