Mathematics - 2018-10-30 (page 3 of 3)

Oskar Tegby

20:01

Thanks, @Ted! That's a great approach.

Eric Silva

hi chat

Oskar Tegby

I kind of like functional analysis.

Daminark

Did someone say functional analysis?

Alessandro Codenotti

functional analysis is great but hard

Eric Silva

@Daminark no get ur ears checked nerd

Oskar Tegby

20:05

Yeah

Daminark

o ok

Oskar Tegby

How does compactness give a minimum for the difference?

@Ted?

Leaky Nun

20:38

@OskarTegby extreme value theorem

MatheinBoulomenos

In a Banach space? Spheres are not compact here

pilotmath

Hello

Suppose random variables X_1, X_2 are independent, then the distribution function of X=(X_1, X_2) determined by the integration limits (which vary depending on the r.v. X) on the joint density(which is simply the product of densities) right?

Semiclassical

that sentence doesn't really parse

at least not in english

Leaky Nun

20:53

@Semiclassical it parses in french

Semiclassical

fair enough

pilotmath

21:11

the distribution of any random variable X of two random variables X_1, X_2 (i.e. X=(X_1, X_2) is integrating the joint density with limits that are in accordance with the relationship between X_1 and X_2 and how they work to form X right? (does that make more sense lol)

Sharath Zotis

Can anyone help me understand open sets in terms of epsilon neigbhorhoods?

For example why is $\mathbb{N}$ not an open set?

pilotmath

@SharathZotis because you can epsilon to be 0.5

Sharath Zotis

$V_{\epsilon}(1)$

pilotmath

and the nbhd would contain rational and irrational numbers

Sharath Zotis

Ok that makes sense

but how about for something like $[1,2)$

pilotmath

21:14

Thus the nbhd cannot lie in \mathbb N

what about it?

Sharath Zotis

so what is the rational here vs (1,2)

Lets look at the epsilon neighborhood for 1

$V _ { \epsilon } ( 1 )$

Crosby

Hello

Sharath Zotis

lets say epsilon is .5 then my epsilon neighborhood $V _ { \epsilon } ( 1 ) = (.5,1.5)$

clearly $(.5,1.5) \notsubseteq [1,2)$

Crosby

Could anyone shed some light on what prerequisites I might need to begin Rudin's Principles?

Sharath Zotis

however couldnt the same be said about (1,2)

pilotmath

21:17

well the point is that for any point in the open set, there exists a nbhd with some radius \epsilon > 0

Sharath Zotis

(.5,1.5) is not a subset of (1,2) either right?

pilotmath

it doens't need to be true for every epsilon, just an epsilon

Sharath Zotis

I see

pilotmath

so you just need to find one epsilon that will give you a nbhd that is contained in the set

Sharath Zotis

then how do we know for sure that in [1,2) we didn't just pick the wrong epsilon?

pilotmath

21:18

I'm not sure what you mean by that question

if I understand you correctly, it's that at point 1, any nbhd of radius epsilon, will contain a point that's not in [1,2)

Sharath Zotis

so for [1,2) there exist no epsilon that allows for the nbhd to be a subset of [1,2)?

ah I see

makes a lot more sense

pilotmath

I suggest reading through the definition of an open set more carefully. But as I've said, the issue with [1,2) is that you take any radius centered at 1, no matter how small you'll contain a point that's not in [1,2) (namely any number strictly less than 0)

Sharath Zotis

for (1,2) what epsilon would that be though?

pilotmath

and open set requires every point to be an interior point

well in (1,2) you dont concern yourself with a nbhd centered at 1

because 1 is not in the set

Sharath Zotis

I see, I will review the definition more thoroughly but you have cleared many of my doubts. Thank You!

pilotmath

21:22

Sure thing!

Two messages above, I meant any number strictly less than 1*

not 0

user330477

I am sorry for asking this again. If we define closure of a set as the smallest closed set containing that set, then why must the closure exist.

I was given this reasoning which implicitly assumes the existence of the closure of a set. "X being the smallest closed set containg Z means that every closed set that contains Z also contains X, you show that there is such a set by exhibiting a concrete construction, i.e. the intersection of all closed sets containing Z, there are not two parts about this"

Leaky Nun

as I said, you're over-complicating the question.

user330477

@LeakyNun You are not understanding my question.

pilotmath

@user330477 are you aware of another definition that says something like the closure is the set + its limit points

If you take that approach, the existence part may be more apparent

MatheinBoulomenos

@user330477 Let Z' be the intersection of all closed sets containing Z, then Z' is contained in every closed set containing Z and it is closed, thus Z' is the smallest closed set containing Z (hence the smallest closed containg Z exists!). At which point do you think that we need to assume existence?

user330477

21:33

@MatheinBoulomenos Can we show that to things are equal without showing the existence of one of them? If yes, then you are right. If no, then we need a different argument.

Mike Miller

That's not a coherent sentence. What did you not show the existence of?

And what are you trying to show are equal?

Leaky Nun

3 hours ago, by user330477

Define the closure of a set Z as the smallest closed set that contains Z. Show that the closure of Z actually exists, and that it is the intersection of all closed sets that contain Z. I am having trouble with understanding showing that closure actually exists? I seems a bit circular if we prove this after proving the second result.

@MikeMiller ^ original question (in case you don't know)

user330477

@MikeMiller The intersection of all closed sets containing $Z$ is actually the smallest the closed containing $Z$.

We need to show existence of closure of $Z$.

Kasmir Khaan

This can be proven via group theory

Hi chat :D

like if we get all subgroups of R that contain Z

MatheinBoulomenos

@user330477 so to give a different context, do you agree that the fact that $1 \leq n$ for all $n \in \Bbb N$ implies that $1$ is the smallest natural number and that we don't need to know a priori that a smallest natural number exists?

Kasmir Khaan

21:36

and intersection of all subgroups is again a subgroup

MatheinBoulomenos

Hi @Kasmir

Kasmir Khaan

@MatheinBoulomenos Hey mathein ;D

@LeakyNun Hey Leaky :D

Leaky Nun

@MatheinBoulomenos triggered

Mike Miller

Yes, and the first part of the sentence is in fact the part that does that. 1) "The intersection of all closed sets containing Z" is closed. 2) "The intersection of all closed sets containing Z" is contained in any closed set containing Z, tautologically.

Leaky Nun

hi @KasmirKhaan

MatheinBoulomenos

21:37

@LeakyNun you can replace $1$ with $0$ if you prefer

Mike Miller

Therefore you have constructed a closed set containing Z that is contained in any other closed set that contains Z.

This is what "smallest closed set containing Z" means.

Kasmir Khaan

in the algebra sense we can go to <1>

but am not sure what context is this, real analysis he is doing ?

Mike Miller

You could argue via Zorn's lemma or something that there is such a smallest closed set containing Z if you want but why bother.

You constructed it by hand. That's better than arguing abstractly.

user282639

It appears to be real analysis

user330477

@MatheinBoulomenos Thank you for your example this does clear things up.

Kasmir Khaan

21:39

why Zorn !

Zorn used to show an infinite process has to end

not smallest element exist no ?

Adam

Kasmir Khaan

@MatheinBoulomenos What do you think of Lang? ever read any of his books ?

user330477

@MikeMiller Thank you

Mike Miller

@Kasmir Zorn has many applications.

Kasmir Khaan

Aha neat =p never seen it used to prove smallest element before

Thanks Mike Miller !

Kas out ! ._.

that means kasmir is out

MatheinBoulomenos

21:42

@KasmirKhaan I use his graduate algebra book as a reference and sometimes to refresh a topic. I also worked through a few of his exercises. It's densely written, but quite comprehensive

I wouldn't recommend it unless you already have a good background in algebra

Kasmir Khaan

@MatheinBoulomenos Okay thanks ! am not using the graduate version yet, doing undergrad one and the lin algebra as repetition =p

from what iv seen so far , his style is very good

user2236

Welcome @HenningMakholm

Kasmir Khaan

like one can follow what he is saying not like other books ><

MatheinBoulomenos

@KasmirKhaan I don't know his other books except for the grad algebra one and his book on modular forms, but he wrote a lot. It's good if you found a book that fits you

Kasmir Khaan

@MatheinBoulomenos his book on graduate algebra is very famous ! and got alot of good reviews from experts, what is modular forms ?

each time i came here mathein, you say something i never heard before ._.'

that still impresses me :D

Daminark

21:47

Lang has a book on modular forms? Holy crap this guy wrote practically everything

MatheinBoulomenos

Why did Bourbaki stop writing books? They got discouraged when they found out that Lang was one person.

Kasmir Khaan

iv seen the list of books its amazing

haha really ?

MatheinBoulomenos

no, that was a joke

Kasmir Khaan

but lang was also bourbaki

aha okay ><

MatheinBoulomenos

modular forms are certain functions of a complex variable that play an important role in number theory.

Mike Miller

21:49

Bourbaki just put out another book though

Rumor says it's not great

Kasmir Khaan

aha neat ._. there is alot of stuff to learn !

user2236

@MikeMiller link please

is Wildberger apart of this "new" Bourbaki?

Leaky Nun

@MikeMiller also link please

Mike Miller

You could Google just as easily as I could :p it's on algebraic topology

Ah, I probably used 'just' poorly above

I mean that in opposition to the era it started in

Instead of as in past few days

My bad

user2236

Thanks, that narrows it down :P

Fargle

21:58

Livre XI: Topologie algébrique

Leaky Nun

that's interesting

(1967). Livre X
(2016). Livre XI

user2236

very interesting

A new generation of Bourbaki

Do they have the same age/membership restriction?

user76284

I have a question regarding integration over manifolds. If I'm trying to integrate the norm of the differential (https://en.wikipedia.org/wiki/Pushforward_(differential)) of a smooth map $f : \mathcal{M} \rightarrow \mathcal{N}$ over $\mathcal{M}$, I have to multiply by the volume form, right? As in
$$\int_\mathcal{M} \|\mathrm{d}f(x)\| |\det \mathrm{d}f(x)| \mathrm{d}x^1 \dots \mathrm{d}x^n$$

Adam

22:15

I thought that communicating an inconsistent set of statements about Euler's theorem would be the iconic shenanigan of our modern times, I thought that this would result in my social acceptance, I don't understand why I am still not popular

Fargle

Euler's theorem: $e^{i\pi\phi(n)} = \pm 1$, where $\phi$ is the totient function

Semiclassical

22:41

@Fargle lol

stronger statement: $e^{i\pi \phi(n)}=1$ almost always

Leaky Nun

23:23

every day I waste my time scrolling through social medias and then complain that I don't have enough time...

7

Semiclassical

23:46

I'd tease you for wasting time on social media, but I waste plenty of time on non-social media

would this room count as social media? I'd tend to say no

2

Mitch

@Semiclassical Wikipedia on social media says yes

Semiclassical

1) Social media are interactive Web 2.0 Internet-based applications.
2) User-generated content, such as text posts or comments, digital photos or videos, and data generated through all online interactions, is the lifeblood of social media.
3) Users create service-specific profiles for the website or app that are designed and maintained by the social media organization.
4) Social media facilitate the development of online social networks by connecting a user's profile with those of other individuals or groups.

Mitch

@user2236 No. Wildberger is on his own.

Semiclassical

hmm, that does pretty well describe the Stack Exchange system as a whole.

Mitch

anything with comment chains or chat

Semiclassical

23:58

the fact that my log in to SE is actually through my Google account is a testament to the last one

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$Mathematics$ Mathematics