Mathematics - 2018-04-04 (page 2 of 5)

Leaky Nun

08:29

3 hours ago, by Zee

And I refuse to read your Latex nonsense

wtf am I reading

why do cranks exist @Secret

Tobias Kildetoft

@LeakyNun That is an interesting question. You should read web.mst.edu/~lmhall/WhatToDoWhenTrisectorComes.pdf

ÍgjøgnumMeg

@TobiasKildetoft May I ask you a quick question? :)

Tobias Kildetoft

@ÍgjøgnumMeg Sure

ÍgjøgnumMeg

@Tobias Cheers! If $I, J$ are ideals in some ring $R$ and $I, J$ are not coprime, why should they be contained in the same maximal ideal? It's a step in a proof in Milne's notes that I'm not quite sure about

Tobias Kildetoft

@ÍgjøgnumMeg The ideal $I+J$ is then not the entire ring, so it is contained in a maximal ideal

and both $I$ and $J$ are contained in this as well then

ÍgjøgnumMeg

08:43

Ahhh okay, that makes sense

Thank you

Tobias Kildetoft

Where the last part follows from the usual Zorn argument on the quotient by $I+J$

skullpatrol

@LeakyNun to try and turn peoples crank :-)

Leaky Nun

@skullpatrol that's the "what for" interpretation of "why", instead of what i intended, i.e. "what is the cause"

skullpatrol

they get enjoyment out of it?

Leaky Nun

who would enjoy being intentionally stupid

skullpatrol

08:47

they want to crank you up, by acting stupid; and then sit back and laugh while they watch you spin

Tobias Kildetoft

@skullpatrol If only that were the case for a lot of them.

Leaky Nun

interesting

Tobias Kildetoft

The article I linked gives a pretty good insight into what sort of thinking lies behind many of them

Leaky Nun

@skullpatrol so you're suggesting that they don't believe what they say?

@TobiasKildetoft reading

@TobiasKildetoft wait it actually works

Tobias Kildetoft

what works?

Leaky Nun

08:53

cut the angle out, roll it up in a right circular cone, and trisect the circular base

Tobias Kildetoft

sure, but that involves more tools than allowed

Leaky Nun

lol

Tobias Kildetoft

Also, can you trisect a circle? (I never really did any of those constructions)

Leaky Nun

yes, because you can find the center, and trisecting a circle = constructing 120 degree

and cos(120) and sin(120) definitely live in the quadratic fields :P

Tobias Kildetoft

right

Leaky Nun

08:57

(the error is that, of course, you can't roll it up in a perfectly circular cone)

Tobias Kildetoft

What sort of mathematical operation would correspond to rolling it up I wonder?

Leaky Nun

constructing a circle whose circumference has the same length as the arc length

(you can't even straighten up the arc length, good luck giving it a different curvature)

(pi is transcendental)

Tobias Kildetoft

Right, and constructing that circle is clearly strictly harder than just trisecting the angle directly

Leaky Nun

right

at least it is algebraic :P

Secret

@LeakyNun because strong beliefs?

Leaky Nun

09:10

hmm

Silent

09:21

@LeakyNun, will you please look at this? It seems really big question, but most of the meat is definition etc, which I think you are well aware.

Leaky Nun

@Silent sorry, not familiar

Silent

@LeakyNun alright.

user193319

12:00

1

Q: Corollary 5 in Royden-Fitzpatrick's Real Analysis: Convergence in Measure

Corollary 5: Let $\{f_n\}$ be a sequence of nonnegative integrable functions on $E$. Then $$\lim_{n \to \infty} \int_E f_n = 0 ~~~~~~(5)$$ if and only if $$f_n \to 0 \mbox{ in measure on } E \mbox{ and } \{f_n\} \mbox{ is uniformly integrable and tight over } E ~~~~~(6)$$ H...

Silent

Suppose $N$ subgroup of $G$, then for all $u,v\in G$, $uN=vN$ if and only if $v^{-1}u\in N$ if and only if $u^{-1}v\in N$

Am i right?

Tobias Kildetoft

@Silent Yes, except you should remove the "all", or the statement is not quite right

Silent

oh, i see.

@TobiasKildetoft Is it true that $u,v\in G$, $uN=vN$ if and only if $uv^{-1}\in N$?

Tobias Kildetoft

no

that would give equality of the right cosets instead

Silent

ok

Slereah

12:37

Are there any non-trivial fiber bundles with base manifold $\mathbb R$?

Mike Miller

No

Slereah

Not too unexpected, but what would be the idea behind the proof?

0celo7

@Slereah $\Bbb R$ is homotopy equivalent to a point and homotopic maps induce isomorphic pullback bundles.

Slereah

thanks

Secret

thinking about essential singularities: Suppose I have a piece of cloth that has the shape of the complex function exp(1/z), how can I figure out what simple shape it is homeomorphic to cause it definitely does not look like it will be homeomorphic to an annulus?

Leaky Nun

12:53

@TobiasKildetoft how so?

0celo7

@Secret what? a graph is homeomorphic to the base

Tobias Kildetoft

@LeakyNun Well, the statement would still be true, but in an uninteresting way, since assuming that $uN = vN$ for all $u,v$ leads to $N = G$.

Leaky Nun

@TobiasKildetoft that's the wrong interpretation

the correct interpretation is $\forall u \forall v [uN=vN \iff u^{-1}v \in N \iff v^{-1}u \in N]$

Tobias Kildetoft

@LeakyNun It is not the intended one, but it is the only one that one should have, since otherwise there is no point in using the "all"

Leaky Nun

whereas you interpreted it as $[\forall u \forall v uN=vN] \iff u^{-1}v\in N$ which makes no sense to begin with

(u is undefined)

@TobiasKildetoft for all natural n, n is even iff n=k+k for some k

"but 1 isn't even!"

that's the whole point behind all

Mike Miller

13:00

@0celo7 That's a little unfair since the proof of that uses the proof for a bundle over X x I. But only a little.

Alternatively pick a complete vector field lying above (d/dt) downstairs and flow on that.

Secret

@0celo7 uh, which kind of base you are referring to? The base of the topology or something else. Cause I don't see how the graph of exp(1/z) is tied to graph theory?

Leaky Nun

@Secret an X-Y graph is homeomorphic to the X-axis

Secret

even when there is some kind of puncture in the form of an essential singularity?

Mike Miller

@Secret if f is a continuous function, then z mapsto (z, f(z)) is a homeomorphism

From the domain of f to its graph

Exp(1/z) has domain the punctured plane

Secret

I always thought the wild oscillation behavior of visiting every complex value infinitely many times near the essential singularity will prevent it from being homeomorphic to the punctured plane?

Leaky Nun

13:04

oh right, your thing has poles

then it is homeomorphic to the punctured plane

Secret

Ok I see

Mike Miller

There's no wild behavior near any point in its domain

Only in a neighborhood of 0. 0 is not in the domain

Saying that it's crazy near 0, since 0 isn't in our domain, is like saying it's crazy "near infty"

It's a growth issue

Secret

hmm...

yeah I think I got it now, the only place where wild behaviour occur is in the neighbourhood of 0, whih is not in the domain, but zooming in to any point near zero, its neighbourhood is nice and continous, thus everything is fine within its domain

Sometimes I get mislead that when things go wild in one neighbourhood, it will mean all the points involved also have wild behaviour in their neighbourhood

It kinda reminds me of that $\frac{2xy(x^2-y^2)}{x^2+y^2}$ counterexample in multivariable calculus, the parabolic shaped ridge where the limit becomes discontinous only occur right along that path and any infinitesimal deviation from that path and you already get a value of zero

I still sometimes get mislead when I plot their graphs, since that degree of resolution cannot be shown on the computer render of the graphs themselves

IP Address

13:33

$a\mathbb{C}b$ is pronounced a choose b

0celo7

@MikeMiller My first thought is to parallel transport along a connection. I think this gives a nonvanishing section.

IP Address

$a\mathbb{P}b$.... here whats $\mathbb{P}$

$\mathbb{C}$ is the binomial coefficient.. but I don't know what P is

0celo7

Here one uses the fact that there’s a complete vector field on the base, as you said.

Er, maybe just that there’s a nonzero section on the tangent bundle of the base.

IP Address

@XanderHenderson good morning to you also :)
$a\mathbb{C}b$ is pronounced a choose b
$a\mathbb{P}b$.... here whats $\mathbb{P}$
$\mathbb{C}$ is the binomial coefficient.. but I don't know what P is

0celo7

P means pick

Permutation

Xander Henderson

13:40

I've never seen them in blackboard bold however

that kind of hurts my eyes... >:(

IP Address

@0celo7 thank u!

Secret

14:03

^^

BAYMAX

What is the best way/trick to write the series of $ln(1+x)$ easily?

Leaky Nun

@BAYMAX as an integral

ln(1+x) = int 1/(1+x) dx

use GS to rewrite 1/(1+x) as 1-x+x^2-x^3+...

\> implying that limits commute with integration

and then integrate each term to get x-x^2/2+x^3/3-x^4/4+...

Semiclassical

Similarly to what Leaky is saying, you want to have $\frac{d}{dx}\ln(1+x)=\frac{1}{1+x}=1-x+x^2-x^3+\cdots$

Leaky Nun

@Semiclassical hey that's similar to what I'm saying

Semiclassical

and since $\ln(1+0)=0$, you can deduce what the proper antiderivative is

@LeakyNun and this statement, in turn, is similar to what I was saying. nice symmetry there :P

It's not really different

I just like framing it in terms of differentiation and term-wise anti-differentiation

Sha Vuklia

14:16

guys, which axiom in ZFC allows us to pick an element from an infinite set?

Leaky Nun

@ShaVuklia finite choice is by logic, not any ZFC axiom

Sha Vuklia

what do you mean by logic?

Leaky Nun

if for a statement p you can prove that ∃xp, then you can have p[x:=c]

i.e. any non-empty set, by definition, has an element

but if you define "infinite", I might be able to pick the element more explicitly

Sha Vuklia

hm, I haven't had much logic yet. maybe I should come back at this after a couple of weeks

Leaky Nun

one definition of infinite is that there is an injective function N->S

where S is our set in question

Sha Vuklia

14:20

right, then you can just pick N(1)

Leaky Nun

right

Sha Vuklia

alright, that makes enough sense for me for now

thanks!

Leaky Nun

but another definition is that it bijects to a proper subeset of itself

user193319

Dumb question: If $P$ and $Q$ are partitions of $[a,b]$, in the sense used in Riemann Integration, then is $P \cup$ Q$ a partition that refines both $P$ and $Q$?

Sha Vuklia

are those definitions equivalent?

Leaky Nun

14:21

yes

so go and take your logic course :P

Sha Vuklia

hahahah, yea it's started this week

actually, I'll be following 2 logic courses at the same time

Leaky Nun

nice

then you'll be doubly logical

Sha Vuklia

an introductory course from last year which I skipped, and a normal course for 2nd years

@LeakyNun lol

Alessandro Codenotti

14:41

What kind of logic are you doing in those courses? @Sha

Akiva Weinberger

@user193319 If by $P$ you mean the set of endpoints of the intervals in the partition, and similarly for $Q$, then yeah

Secret

I am going to draw a picture on some unordered set later, just because I am kinda bored

user193319

@AkivaWeinberger Yes. That's what I mean. Thanks!

Sha Vuklia

@AlessandroCodenotti looking at the index of our syllabus of the introductory course, we will do syntax/semantics/natural deduction for proposition logic and predicate logic

and the second course, as you might remember, we did Stone duality, and now we're doing everything you would expect from a logic course:p

Alessandro Codenotti

14:56

Is there some model theory too in the course? Löwenheim-Skolem and so on

Sha Vuklia

this is from a final exam

@AlessandroCodenotti yep

Alessandro Codenotti

@ShaVuklia I like the third and fifth exercise there

Are you going to cover ultraproducts too? Looks like it from the 5th exercise

Sha Vuklia

lol, it seems like it

Alessandro Codenotti

LoL i didn't read 5c, only 5b, they're mentiobed explicitely later... derp

Sha Vuklia

oh hahaha

Leaky Nun

15:00

@AlessandroCodenotti well 3 is quite standard

Sha Vuklia

anyhow, I had a question @general: Munkres says that a subspace $A\subset\mathbb R^n$ is compact iff it is closed and bounded in the euclidean metric $d$ of the square metric $\rho$. Could they have also just written that $A$ is compact iff closed and bounded in any metric, because metrics are equivalent anyways?

Alessandro Codenotti

@LeakyNun indeed, but it's an important standard exercise

user21820

← 3 messages moved from CRUDE

Leaky Nun

@ShaVuklia 4 will let you practise formal proofs

Sha Vuklia

yea, I need that:p

user21820

15:02

@IPAddress $\mathbb{C}$ is pronounced "complex numbers". You don't use that symbol for "a choose b".

Alessandro Codenotti

@ShaVuklia not all metrics on $\Bbb R^n$ induce the same topology

Sha Vuklia

o wait, maybe I'm confusing the equivalence of norms with something nonexistent

user21820

And technically the more standard notation is C(n,k) for "n choose k" and P(n,k) for "n pick k".

Leaky Nun

@ShaVuklia should I introduce that to you or do you prefer to wait until they teach it

Sha Vuklia

equivalence of norms? @Leaky

Leaky Nun

15:03

no, proofs

Sha Vuklia

@AlessandroCodenotti o yea, that's true

oh, well if you want to, I wouldn't mind, but today I have a lot of topology and graph theory to do:<

Leaky Nun

ok

oh and how long is your course?

Sha Vuklia

if you mean the logic one, then it's 4 months

Leaky Nun

when is your exam?

Sha Vuklia

technically in 8 weeks, but I will skip it, so it will be in 13 weeks for me:p

it's an elective course, so I'll focus on other courses first, and then just go for the retake for this one

Leaky Nun

15:05

how does a 4-month course have an exam in 8 weeks

Sha Vuklia

lol, what do you think:p

Leaky Nun

no idea

user193319

If $f : I \to \Bbb{R}$ is continuous, where $I$ is a compact interval, must $f$ also be Lipschitz continuous?

Sha Vuklia

we're already half way

Leaky Nun

oh.

45 mins ago, by Sha Vuklia

hahahah, yea it's started this week

but this?

Sha Vuklia

15:07

I meant the introductory course

that one is a 2 month course

Leaky Nun

never mind

Sha Vuklia

:p

Leaky Nun

@user193319 $x\sin(1/x)$

Kasmir Khaan

@LeakyNun hi leaky :D

I have a non math question ._.

Leaky Nun

just ask

Kasmir Khaan

15:13

is there a program that records what i do on computer?

someone gonna show me how to use program

and i need to see it few times

Leaky Nun

no idea

Kasmir Khaan

:(((((((((

thanks anyway leaky :D

ill google

again

Secret

bandicam records your screen as a video, but I suspect you need something more

Kasmir Khaan

no just that

record screen

no need for sound

Abcd

0

Q: Why do the professional mathematicians believe blindly in so meaningless concepts as Infinity?

To refute such a concept as Infinity (or many infinities) in mathematics doesn't at all require all that big efforts mainly from its own definition in mathematics. To explain this very simple fiction in human minds, just consider the natural numbers, where simply they are a continuous chain of "...

logic ethics philosophy-of-science metaphysics philosophy-of-mathematics

Kasmir Khaan

15:18

thanks secret guy :D ill check that and keep digging

Semiclassical

@Abcd As a rule of thumb, I don't expect a lot of good things to come from questions which start from the premise that they're obviously right and all the experts are obviously wrong

2

Silent

@LeakyNun, Why equality holds here but need not hold here?

Leaky Nun

@Silent because $\sup_{\theta \in \Theta} (f(\theta)+g(\theta)) := \sup \{f(\theta)+g(\theta) \mid \theta \in \Theta\}$

$\{f(\theta)+g(\theta) \mid \theta \in \Theta\} \ne \{f(\theta) \mid \theta \in \Theta\} + \{g(\theta) \mid \theta \in \Theta\}$

Silent

ok, thank you.

Secret

QED

Rule of thumb. whenever people of that sort is identified, force them to read an academic journal on identity politics and watch as you manically laughing as their mind get contorted and eventually dissolved into a mush by the nonsensical jargon

PS: I like to torture people from the inside

3

Semiclassical

15:28

Along similar lines: convince them to go into lit crit, where their ability to come up with such imaginary worlds of argument would probably give them a good career :P

Secret

indeed indeed

Semiclassical

If you think this is too hard on literary criticism, read the Wikipedia article on deconstruction.

is the inspiration for that

This is also why I get annoyed at crackpot nonsense, since it becomes clear that they're imposters not only to us but to themselves

Lozansky

It would be the reversed for me :P

Semiclassical

they've convinced themselves that they're experts

Secret

I probably will survive a lit. crit indefinitely by even mumbling word salad

I usually will talk to a crackpot until they reached a point to promote their pet theories again

after that I stop interacting with them

until the next cycle happens

Semiclassical

15:32

right.

The approach I'd love to be able to take with crackpots is to act like I'm agreeing with them and lead them unknowingly into contradiction

but that requires far more patience than I actually have in practice

Secret

Meanwhile, a lot of my questions are sharp to the point of making professors uncomfortable and they are designed in a way so that a (sufficiently rational) crackpot will at least be put into a state to critically assess their pet universes

which is one reason I ignore the emotional portion of their response completely and just focus onto the point they need to explain themselves

it might end up dissolving their pet universe in some way, or it my strengthening them or changing them

but at least they are not brain dead and are thinking. thus wish them best of their luck getting out of the labyrithn of their delusions

I think I have one successful record of getting a mainstream idea (quite significantly watered down) across to JD

Semiclassical

Psychologically, one could probably argue that my dislike for crackpots is a matter of projection: I get exercised by their delusional logic because I know that I get trapped in my own bad logic

Secret

I do sometimes worry I am not skillful enough to tell the crack from the mainstream, especially myself have a lot of crazy ideas that blurs the boundaries easily and causing me to falsely see sense in their crackpot responses

I always felt like I am constantly at war with my pet theory self, which is why I need the community to keep that in check, besides my own readings

Eric Silva

@Semiclassical literary criticism isn't as bad as the joke makes it out to be but I think the view that it is that bad is kind of normalized amongst stem people and it makes me sad

Akiva Weinberger

Heyo

Semiclassical

15:39

yeah, i guess it's sorta the equivalent of the non-stem view that math equals arithmetic

Secret

@EricSilva well tbh, I actually knew nothing about what those guys do, thus technically I have no comment

Fargle

@Semiclassical Somebody likes Plato.

Semiclassical

lol

Eric Silva

Literary critique stuff is often super opaque and hard to approach so it gives the impression of being gibberish but math has the same complaint I guess

Semiclassical

Math at a certain level, at least.

When you can do direct calculations and such, it's a bit easier to evade that complaint

Secret

15:41

Well, general topology is pretty opaque to first learners, as well certain types of abstract algebra, where there are nested jargons

Eric Silva

it's opaque for different reasons is the thing

i think it's like more obvious why stem subjects might be opaque, they're technical, but it's perhaps not so obvious why softer subjects might be

Secret

As for identity politics, there's an article mentioning how it has gone so opaque that even insiders get confused on what is happening and they just parroting the jargon everywhere without understanding why

truthdig.com/articles/…

Semiclassical

I think that's probably an exaggeration as well

Eric Silva

intersectionality is not a complicated idea

Akiva Weinberger

I find identity politics boring, as a subject

Semiclassical

15:46

it depends, for me

if we're talking about identity politics in the sense of "what circumstances frame the experiences and opportunities of group X", that's fine

Eric Silva

@Semiclassical tfw m a t e r i a l c o n d i t i o n s marxism intensifies

Semiclassical

right

material conditions matter

but it has a tendency to devolve down to a form of tribal warfare

Akiva Weinberger

@Semiclassical I guess that makes sense

Semiclassical

and when identity politics devolves into tribal politics then it becomes pretty problematic

Akiva Weinberger

"My history class is a struggle… but all history is class struggle."

Eric Silva

15:52

@Semiclassical i think i said this when we were in the hbar but i think that the tribalism isnt what's wrong (in fact i don't think tribalism is a priori a bad thing but im maybe a marxist) i think it's the priorities and performativity that's bad

or maybe it's that the tribes arent organized right

Akiva Weinberger

Is economics just trying to answer the question, "How do we make people want to make good things?"?

How do we make people want to make systems that make people want to make good things?

Eric Silva

maybe but in practice economists are studying the ways the capitalist mode of production organize

Semiclassical

@EricSilva I think we differ on that, yeah

Akiva Weinberger

Every system creates incentives, so I guess the problem is to make systems that make good incentives

Adeek

hey @AkivaWeinberger @EricSilva

Akiva Weinberger

15:56

That's my opinion on justice as well. A set of laws are just if they properly incentivize good behavior

Adeek

do you guys know if there is any geometric intuition hidden in the coeffients of series expansion of holomorphic functions ?

Akiva Weinberger

They have to go to zero relatively quickly, depending on the radius of convergence, or something like that

I don't remember the details

Semiclassical

to put it in kantian terms, it seems like any argument which is basically 'tribal' is always going to end up being a hypothetical imperative rather than a categorical one

Eric Silva

is it?

Adeek

yeah I mean to derive it algebraically is easy

but if there is some geometry

Semiclassical

15:59

Well, let me try to say it like this.

Eric Silva

@Semiclassical class struggle is by nature tribalistic and i think the needs of one class is motivated by a categorical imperative while the other is motivated by perpetuating exploitation

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$Mathematics$ Mathematics