Mathematics - 2022-07-18

Calvin Khor

01:36

@AkivaWeinberger $\overset{\displaystyle\triangle}{\triangle\triangle}$

Akiva Weinberger

@CalvinKhor Nice TeX'ing

The question asks if that's the only way

Calvin Khor

yeah i read. I just like pictures

shintuku

that answer is simply too powerful

it won't do, something tamer is needed

hansika

02:37

Hi! Is there any not-so-tedious way to solve it?

Ted Shifrin

03:04

I don’t even understand the question. All four lines are concurrent?

polite proofs

03:31

@TedShifrin Hello Dr. Shifrin

What's a good site to upload a PDF on?

I would like to share my progress with the solutions manual for your homework exercises

hansika

03:55

@TedShifrin yes, all the 4 normals at the extremities of these chords are concurrent, that's what i could interpret, altho i am not really sure.

Calvin Khor

@politeproofs i recently found out that you can upload pdfs to archive.org.

Ted Shifrin

Well, I know a criterion (using easy linear algebra) for three lines to be concurrent. But I don’t see a good approach yet.

@politeproofs What?

hansika

@TedShifrin i have a solution for it which i couldn't understand. shall i share?

Ted Shifrin

If you want.

hansika

Ted Shifrin

04:05

The first sentence makes no sense. The second and third are fine.

Fourth doesn’t make sense either …

hansika

ohh..i think both the question and the solution are a bit vague

Ted Shifrin

I’m not sure it’s right.

The first equation is satisfied by the 4 intersection points.

hansika

@TedShifrin intersection points of the given lines and the ellipse?

Ted Shifrin

Yes

Nothing to do with normal lines concurrent.

Ah, but the rest does work now.

Those four points satisfy equation 2.

hansika

ohh yess..and those 4 points satisfy the above curve, so both are same curves?

Ted Shifrin

04:17

Right.

hansika

okayy

also could you please tell me that the 1st equation type is valid for every conic? like if we have to find a curve for the intersection of two lines and a conic we can use equation of similar type?

i knew it only for a circle

Ted Shifrin

True for any curve at all!

hansika

okayy, thank you for your time! :)

Ted Shifrin

Sure.

Balarka Sen

09:38

@LeakyNun i mostly dont read academic philosophy because i cant understand what they're saying. soft, literary philosophy i enjoy.

dostoyevsky is good

Leaky Nun

09:55

@BalarkaSen could you give me like a brief introduction of what his philosophy is about

shintuku

10:49

@LeakyNun it's the middle of the 19th century

people keep making damn revolutions, it's crazy

they're about as far away from the french revolution as we are from ww2, and the 1848 revolutions set the entirety of europe on fire again, like just 15 years ago

as far from the french revolution as we are from ww2, you say? that is, people started thinking (period) about 50-80 years ago

people keep asking for monarchs to just die already, keep talking about "the people" and whatever

pfft, weren't those guys supposed to work at the fields and in the factories, how the hell are they supposed to run a government

(the bourgeois here get mad because they get grouped with low-life proletariat)

you wonder, ok, well, people used to have a clear idea of what's up. hell the conservative elite still has a clear idea of what to do

the world is the world of god, follow the bible, obey authority, whatever

the bourgeois elite too, has an idea of what to do. freedom or whatever, Man, Humanity, Democracy

big words for idiots with barely any parliamentary representation in most of europe (wait where do parliaments even exist yet)

well here's you. you, my dear common liberal bourgeois child of the 1880s, your parents are probably doctors or own a factory or something, and you're probably not part of the conservative elite clinging to power

france just brushed against proto-fascism with Boulanger

germany's got bismarck fighting the socialists (christ we had liberals and conservatives, now what the ef is a socialite?) and the liberals at the same time, doing a good job but why are there so many different ways of thinking

Akiva Weinberger

Wait are you saying his philosophy was "everything went to sh-t when the masses took power and invented democracy"?

Was he a fascist?

shintuku

austria-hungary seems to have a weird idea of the state, how are you supposed to make a state with two ethnically different populations, i thought those french guys said we needed a nation-state?

btw austria-hungary is huge

2 sec akiva i'm rewriting history

britain is doing surprisingly good. gotta say, enslaving most of the world does wonders for political stability

Akiva Weinberger

Making a state with two ethnicities is fine as long as the borders weren't created by the Brits

shintuku

no wonder austria-hungary is still a thing today akiva

russia, oh yeah i forgot to say, you're russian, just emancipated the serfs. your conservative uncle says this'll just make them vagabonds that never work and will get them hooked on whatever is tendance these days (socialism)

Akiva Weinberger

Well I mean Belgium and India are things

shintuku

10:58

hope they won't attempt to overthrow the government, those silly serfs, haha

belgium exists because france and germany needed to not be touching each other, it's integrity guaranteed by the threat of annihilating war between neighboring states that would menace it's existence

india is cool, but we don't know what it would have done to face modernity because of the british assholes

Akiva Weinberger

Oh, I'm dumb, why did I include India in the "Britain not involved" list

I blame sleep-deprivation

shintuku

also, back to dostoyevski, and you, liberal bourgeois child of russia (1% among 99% of an enslaved poor population) you're in russia so you're about to get the 1905 revolution in a few, these idiotic pathetic masses of human skin keep running against the imperial guard to get mowed down, shoulda never emancipated them

insatisfaction brewing, a feeling of change in the air, certainly the Church and God aren't doing their jobs

or maybe God himself wants this? would someone please clarify

you wonder, well ok pause

these are all smart people, i guess, running governments, factories, thinking about the fate of the world

why is there no damn clear winner in all of this

a simple solution

why can't you just trust someone to do their god damn job and run europe

most of all

what the hell are you supposed to do, when all social institutions seem at most transitorily stable and clearly bound to fail and fall

you used to look to europe to give russia an idea of what it could be, but it's been a couple of years now where it's barely less of a shithole than russia

who knows what would happen if daddy bismarck wasn't there fixing everything (spoiler alert: ww1)

also, the great Motherland russia just lost the crimean war

can't really trust your own government to protect itself from exterior threats. blink of an eye, and we could be talking about German Russia? or British Rus? god, such anxiety

GOD

lads, why is this all happening? what am I to do? who do I follow?

it feels like whatever i do, I'll be wrong

so you try consulting with god

but you know, you're having a hard time praying seriously because half the european population thinks your part of a collective delusional paranoia (-freud, in like 10 years from wherever you are now)

and you know there are some pretty smart lads in that population

btw we're starting to have social thinkers like Marx and Auguste Comte with his silly social physics, saying we're the products of our environment or whatever

you thought, you silly russian liberal bourgeois of the 1880s

that everyone's own faith and future is in their own hands (or if your parents went to church, in god's)

and so you succumb to despair because you don't know what to do, or how to be responsible for your actions in a world that's becoming so complex, ambiguous, difficult to naviagte, stealing the remnants of your ability to even be responsible for your own acts, and stealing the ultimate repository of stability and peace that was god, a source of tranquility immune to war and politics

so, what do you do, to find a footing in this world?

the end of a short intro to dostoyevski

Akiva Weinberger

11:14

So OK math question. I have an mxn rectangle. Two players take turns placing down diameter-1 coins that don't overlap with previously placed coins. Whoever cannot place a coin loses. Who wins?

shintuku

i went to sleep thinking about the triangle into congruent fourths question yesterday

this new question is also fun looking

@AkivaWeinberger can you not simplify the problem by making a 1xk rectangle, and supposing they just add one coin after another in a consecutive fashion?

filling up the rectangle going from one side to the other

oh, we're not supposing a grid disposition of the coins in the mxn rectangle

Leaky Nun

@shintuku thanks!

shintuku

np!

Akiva Weinberger

Maybe there's some sort of strategy-stealing argument

shintuku

do they need to add coins in such a way that they touch each other?

would it be possible that you can solve this problem by determining whether there is an odd or even number of coins in a particular end tiling (the resulting tiling from a finished game)?

Akiva Weinberger

11:26

0

Q: Who wins in this simple coin-placing game?

We have an $m\times n$ rectangle. Two players take turns placing down diameter-$1$ coins that don't overlap with previously placed coins. Whoever cannot place a coin loses. Who wins? (Note: coins are not required to fit neatly on a grid, so most turns will have uncountably many options.) I'm fair...

Arright, it's up

I mean the final result can be pretty messy

shintuku

we could first determine whether there is necessarily a definite winner by determining whether there exist at least a tiling with an odd number of coins, and one with an even number of coins

if no such tiling pair exists, then there is a clear winner for every game

Akiva Weinberger

I feel like it does exist

Take that image, remove one of the circles, and shuffle the remaining ones so that you can't put it back in

Should be doable

shintuku

right, that makes sense

seems like you could just remove one and move a single one

it also seems like there is an not too difficult to calculate lower and upper bound on the max number of coins on the rectangle

maybe from there it would be possible to identify conditions of disposition of tilings that would lead to each max number of coins

e.g., given three coins whose centers are at least x units apart from each other, we are guaranteed to achieve the minimal tiling

porridgemathematics

does anyone know if this is true and how to see it? If $t \rightarrow \mu_{n,t}(E)$ is measurable, as a map $D \rightarrow \mathbb{R}$ where $D$ is some planar domain , and $\mu_{n,t} \rightarrow ^{\ast} \mu_t$ in $n$ is a sequence of probability measures supported compactly in $\overline{D}$ which is compact, then $t \rightarrow \mu_t(E)$ is measurable

shintuku

or, it becomes impossible to achieve the maximal tiling

VintageMind

11:39

Hello! Does anyone have like a list of online resources for mathematics that will be helpful for self study?

Akiva Weinberger

OH

@shintuku Argh

VintageMind

I am in high school btw

Akiva Weinberger

First person plays in the center. Forever after that, first person plays directly opposite second player

porridgemathematics

here $E \subset \overline{D}$ is some fixed borel set

Akiva Weinberger

@VintageMind If you know calculus and complex numbers I recommend the YouTube channel zetamath

Otherwise, 3Blue1Brown's "Essence of calculus" series

shintuku

11:41

@AkivaWeinberger haha, yeah that seems like a dominant strategy

porridgemathematics

the convergence is weak$^\ast$ convergence

shintuku

seems like you can do a lot of math for nondominant strategies

VintageMind

@AkivaWeinberger thx will look into it.

Akiva Weinberger

@VintageMind Beyond that, it really depends on what you want to learn. For textbooks, there are several standard piracy sites that will get you PDFs of textbooks online for free

(Don't worry about ethics; textbook publishing is a scam and the authors don't benefit either way)

VintageMind

@AkivaWeinberger I do agree with you on that point, well I mainly suck at geometry which is something I need to improve on immensely

Akiva Weinberger

11:45

Apparently people liked Euclid's Elements for quite a while, though I never went through it myself

porridgemathematics

Im asking because there is this fact that keeps being used in a text im reading, basically that the formula $v(E) = \int \mu_t(E) d\rho$ holds , where $v, \rho$ are some measures (probability measures) and $\mu_t$ is the weak$^\ast$ limit of some measures $\mu_n,t$ in $n$, and $t \rightarrow \mu_n,t(E)$ happens to be measurable

but the author doesnt say anything about why $t \rightarrow \mu_t(E)$ is even measurable, so i cant get past this 'formula' , assuming it is the formula does hold

VintageMind

@AkivaWeinberger Euclid's Elements seems quite nice for starters actually. For me algebra, combinatorics, discrete maths, and all, are fine with me. But geometry, I fall flat on my face for some reason. Guess I lack spacial capability????

Akiva Weinberger

Honestly I was never all that good with geometry either

I mean as long as you memorize a few results (inscribed angle, power of a point) you're probably good?

I dunno, I'm not the person to ask

@VintageMind Generally I'd push back against the "special capability" stuff

It's all down to foundation, practice, experience, etc

The usual

Every saint has a past, every sinner has a future

VintageMind

@AkivaWeinberger Well I mean about "special capability", it seems to me that society really makes a student feel that they need "special capabilities". Like I myself suck in national math olympiads and informatics olympiads and that's depressing.
It shows that despite studying, memorizing, and revising doesn't get me too far. Like there are so many smarter people than me. And they are just in high school just like me.

But of course, I try to see things positively and improve myself daily. Even if my pace is slower than others in some topics, I keep working harding even though I don't get immediate results.

Math is beautiful :)

Akiva Weinberger

12:02

@VintageMind Contest math is very different from applied math is very different from research math…

It's entirely possible to suck at one and be very good at another

user4539917

Beauty for beauty's sake.

Akiva Weinberger

so don't sweat not being able to do Olympiads. It probably won't matter for what you want to do next in math

Akiva Weinberger

12:12

@VintageMind You can also check these out for group theory

youtube.com/playlist?list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv

(Based on the textbook Visual Group Theory)

Akiva Weinberger

12:33

♫ It's my polytope, I can CRY if I want to ♫

(Google says it's the Chan–Robbins–Yuen polytope)

(and also the convex hull of the nxn permutation matrices?)

VintageMind

@AkivaWeinberger Thank you for your advice! Well good night (10:32pm)!

Akiva Weinberger

Night!

user4539917

cya

Balarka Sen

14:44

@LeakyNun how (not necessarily religious) faith prevails as a guiding light in face of the cripplingly futile that is life

simple statement, good novels

@AkivaWeinberger dostoyevsky was an out and out christian and anti-communist, but not a fascist. he freaked out because anarchonihilists were stirring up trouble in college campuses lol

leslie townes

boo

Balarka Sen

@shintuku nice rant

hi @leslietownes

leslie townes

howdy

shintuku

tnx i try to channel my math insufficiencies

Balarka Sen

math is boring

sometimes i wish i studied literature

shintuku

14:55

you can get the juice out of literature by knowing how to do academic research and reading on your own time

read the canon, look up articles on what you find interesting

Balarka Sen

true, but time is the important part there

shintuku

yeah heheh

Balarka Sen

nobody ever has time

shintuku

word

Balarka Sen

do you know anything about atemporality lol

shintuku

14:56

leibniz tried to prove god is needed to sustain continuous time because there's no way to slice time into discrete units

Balarka Sen

hilarious

shintuku

i know right

Balarka Sen

@BalarkaSen i guess what i really mean to ask is if you know who nick land is

lol

shintuku

no unfortunately, from what i've heard he's cray cray

Balarka Sen

hes the shit

shintuku

14:59

what's he on

onto*

Balarka Sen

he thinks that you can print artificial time

just do some organ printing, print multiple brains and put them in parallel computation

2

shintuku

huh

the sorta guy you want at parties

Balarka Sen

yes

hes an accelerationist, he started out as a weirdoleftist and then accelerated to the other end of the political spectrum

now he calls himself hyperracist

leslie townes

shin makes a good point. you can get into literature more or less anytime, anywhere. in math it helps to have access to specialists. and in lab sciences you basically can't do anything without specialists.

balarka: sounds delightful.

i guess it saves other people the trouble of finding a word for it. how considerate

Balarka Sen

lol

geocalc33

15:35

the $f_X$ marginal distribution is the same as the projection of the multivariate pdf onto the first coordinate assuming independence of $X$ and $Y$

geocalc33

16:24

does $L=(0,1)^3$ admit a nowhere zero vector field?

geocalc33

16:42

it can be shown that every noncompact connected smooth manifold admits a Lorentzian metric

Given a dimension 3 noncompact connected smooth Lorentzian manifold, $L=(0,1)^3$ how would you derive the nontrivial nowhere zero vector fields?

Leaky Nun

@BalarkaSen do you have any thoughts about Buddhist philosophy?

geocalc33

I think the trivial ones would be the constant vector fields where they vanish outside the region

so these vector fields will have planes as integral surface solutions

assuming co-dimension 1 foliations and modding the trivial cases out, one is left with c0-dim. 1 foliations whose leaves converge at two points. These are less trivial, but there are some more nontrivial cases. Co-dim. 1 foliations whose leaves converge at 2 points and the topology of whose leaves are spherical in their geometry, i.e. you can make a closed loop around them

Not Tfue

17:07

I don't understand how determinant multiplied $-a_2$ is same as last one.

What identity was used?

I am new to linear algebra so this question might sound dumb.

Ted Shifrin

What is your precise question, @Not?

Not Tfue

:61610785

Why is this true?

Ted Shifrin

So the $a_2$ is not relevant. You can just calculate the determinants and see they're the same. Whenever you add a multiple of one column to another, the determinant doesn't change.

Balarka Sen

@LeakyNun I know very little about this, except reading the Jātakas when I was a kid.

I know some interesting people from the West who integrated Buddhism in their writings. Jack Kerouac is one.

Not Tfue

@TedShifrin right

Ted Shifrin

17:21

So you're OK?

Not Tfue

i think i really asked a dumb question now

@TedShifrin yup

i was thinking if a_2det(A) and a_3det(B) had relationship because they had same column turns out i was wrong

Ted Shifrin

I don't quite follow, but OK.

Balarka Sen

17:51

@Thorgott Take the standard torus (or any other embedding isotopic to this one) in $\Bbb R^3$. Then the $(1, 1)$-curve (or any curve isotopic to it in the torus) has a tubular neighborhood homeomorphic to a Mobius strip with a full twist.

full twist = two 1/2 twists

Akiva Weinberger

18:11

@BalarkaSen Here's a neat fact

If I have a hexagonal prism, I can ask you how many edge-face pairs there are (edge on the face)

(which is 36)

We can represent that by saying $1,2$ corresponds to $36$

I can ask for vertex-face pairs ($0,2$) which is also $36$

and vertex-edge-face trios ($0,1,2$) which is, uh, $72$ I think

All in all we can collate this into an eight-dimensional vector (a list of eight quantities):

$(\emptyset,1;0,12;1,18;2,8;01,36;02,36;12,36;012,72)$

where $012,72$ means there are 72 vertex-edge-face trios (the vertex is on the edge is on the face)

or, without the labels, this is just the vector $(1,12,18,8,36,36,36,72)$

So in general, for a $d$-dimensional polytope, this is a vector in $\Bbb R^{2^d}$, yeah?

I want you to consider the vectors you get from all $d$-dimensional polytopes, for fixed $d$

and consider their span

Theorem: that span has dimension $F_{d+1}$, the $d+1$-th Fibonacci number

Balarka Sen

Why is this interesting

Akiva Weinberger

I guess it tells you exactly how many linear relations you get amongst these values ($2^d-F_{d+1}$)

I just thought it was neat that the Fibonacci numbers showed up there

I didn't expect them to

Balarka Sen

Hm, OK

polite proofs

19:48

@TedShifrin Why did you write what?

Jakobian

21:08

@politeproofs I think google drive would work, or sharing it on overleaf (I'm not sure if that works)

if you're writing on overleaf

polite proofs

@Jakobian No I am not, I guess gdrive is the best option

Jakobian

@TedShifrin This sequence doesn't have a positive limit point.

The strategy is to tend towards the limit point, gradually allowing the possibly bigger/smaller than necessary terms

@Koro What they mean is that there exists a unique order isomorphism from $\mathbb{N}\setminus f(\mathbb{N})$ to $\mathbb{N}$. This has to do with the fact that $\mathbb{N}$ is a well-ordered set.

It can be written as $n_1 < n_2 < n_3 < ...$, and the isomorphism is just a more formal (proper?) version of this

There's nothing scary here if you understand some basic set theory

I think the solution is pretty close to what I had in mind

just more formalized

Balarka Sen

22:20

@AkivaWeinberger Give me a clean proof of the following. Let $f : T^2 \to T^2$ be the $+1$ Dehn twist about a meridian $m \subset T^2$. Then the mapping torus $T(f)$ is obtained from $T^2 \times S^1$ by a $-1$ Dehn surgery on $m \times \mathrm{pt}$.

I think I can see it but I would like to see it better.

Mathematical Emergency

22:31

Define for me, what is math?

I think math is a list of definitions theorems and proofs that describe physical or higher dimensional "structures". Okay now define for me what a structure is.

And it keeps on going ad infinitum. Lol

Akiva Weinberger

You know how computer scientists like data structures?

Mathematical Emergency

@AkivaWeinberger they hate them actually, they're forced by themselves to work problems, it's a drug: STEM

Akiva Weinberger

I was gonna say mathematical structures are like data structures except you don't care about efficiency and they're allowed to be infinitely large

Mathematical Emergency

@AkivaWeinberger that's brilliant

I really like that one

Define care, lol

Damn, that weed got to me, I'm thinking to deeply about our existence

It's a new strain that I'm not used to

Akiva Weinberger

@BalarkaSen Consider a torus outside of a thingy and a torus inside of the thingy

and slide them into each other

or something

I dunno

Mathematical Emergency

22:39

math it up

Convert everything into a ring or a module, take measurements of dimensions

user10478

22:53

Is there an intuitive explanation for when one should care about independence and when one should care about uncorrelatedness? I know the formulas but I always confuse them in the wild.

Jakobian

23:17

@MathematicalEmergency Anything that feels like math

What I said might sound funny, but it's how we humans perceive things. We look at simple cases and derive more complex ones. Anything that gets bound together by some similarity process is covered by the same definition

At least in math case and couple of others

polite proofs

23:44

Dr. Shifrin, could you review the solutions manual I've written so far?

shintuku

ted give us the solutions manual and let's call it quits

polite proofs

drive.google.com/file/d/1krVmntZlQKTRAXZv0J5J92XnNYCu6eAC/…

The formatting is far from final, and also the problem highlighted in red is unfinished, but that's only because writing this took considerably more time than solving it. I included random challenge exercises which didn't look like they would take too long to write up. In the later chapters, I'll try to do all of the assigned homework in the file.

Transcript for

$Mathematics$ Mathematics