Mathematics - 2016-12-02 (page 7 of 11)

Axoren

08:00

@Jasch1 $f(c) = \sqrt c$ for the set of perfect squares

Jasch1

Yea and that returns 0

Alessandro

If you have vectors with 9 entries it doesn't matter whether you write them as row vectors or column vectors, you only need to keep the components in the right order, do you agree? @Steve

Axoren

The density is zero because there are so few of them compared to how many natural numbers there are.

Essentially, they're on two different orders of magnitude.

Steve

@Alessandro yes

Jasch1

Its the same rules for both odds and evens, the sign of infinity is equal to the sign of the leading coefficient

I understand this well actually

Time to start writing

Axoren

08:01

@Jasch1 Good luck

Jasch1

Onyl around 2:30 left to write a Macbeth paper!

Brody

Have fun! Or just, good luck :)

Axoren

Maybe once you're done with this, you won't be able to wash the Math from your hands.

Brody

I'm still iffy about the $f(c)$ thing @Axoren

Not as a snarky doubt, just out of my incompetence

Axoren

@Brody Probably not, the way I described it uses a lot of nuance which takes a bit to realize it's equivalent.

Brody

08:04

Is it formally provable? @Axoren

Jasch1

@Axoren Is the formula we were using for the natural density the same as another formula?

!!google "natural density" formula

Axoren

@Brody Definitely, but requires more effort than I'm willing to do without cookies.

Brody

Do internet cookies count?

Jasch1

is there another formula name that I could search to get the same formula

Axoren

@Jasch1 They are equivalent. They are exactly the same if you count the numbers in order from $0 \to$

Jasch1

08:05

because natural density formula only returns hair products

Axoren

@Brody I'm going to hunt around in the kitchen. If not, I might bake some.

At 3am

Jasch1

is there another formula that is the same that I could find a pic of

Brody

$\displaystyle\lim_{c\to\infty}\dfrac{|\{1,2,\ldots, c\}\cap S|}{c}$

Axoren

^

Jasch1

k

but that looks copied

because I have no idea what the fuck any of that means

Axoren

08:06

Again, it's equivalent to the one I gave you if you count them in order.

Jasch1

i gotta find that

Axoren

Find what?

Brody

@Jasch1 It really looks way too scary for how simple it is conceptually

Jasch1

nah but it looks like i just copied it, and most of my classmates wont understand this

Axoren

$\displaystyle\lim_{c\to\infty}\dfrac{f(c)}{c}$

Brody

08:07

$S$ is the subset we're looking at it, e.g. the set of perfect squares

Axoren

Then give a formal definition of $f(c)$

And explain what a limit is

Jasch1

i need apng of the formula tho

Brody

@Jasch1 Hmm, I think a bright high school student could grapple that no problem

Jasch1

but im lazy

Axoren

Mathematics is not rocket science, but rocket science is mathematics.

Brody

08:09

You don't have to worry about that formula @Jasch1

Axoren

@Jasch1 Write the formula in ShareLatex, render it, and then screen capture it

Brody

Anyways, go write that paper!

Axoren

PrintScreen on your keyboard, Ctrl+V into MS Paint

Then, select the equation with a Rectangle Select tool and Crop it

Brody

Guilty confession... Sometimes, I keep an "Ask a Question" tab open to test LaTeX commands.

Fargle

@Brody That's what ShareLatex is for tho...

Axoren

08:11

@Jasch1 Use the PDF Zoom In feature

Jasch1

Lmao i just saw that

Axoren

Also, don't be embarrassed about asking silly questions.

I'm pretty sure I've asked much sillier and felt terribly foolish

Brody

@Fargle I didn't know about ShareLatex before this, and staying on MSE seemed more convenient lol

Jasch1

your formula looks different in latex

Fargle

@Brody I suppose that's fair. I just always have a tab open whenever I open my browser.

Jasch1

08:12

in sharetex

no divided by

Alessandro

so, @steve, forget about matrix multiplication or any additional structure, as long as addition and scalar multiplication are considered, is there any difference between $\begin{pmatrix} a & b \\ c & d \end{pmatrix}+\begin{pmatrix} e & f \\ g & h \end{pmatrix}$ and $(a\:b\:c\:d)+(e\:f\:g\:h)$?

Axoren

@Jasch1 MathJax uses a different display style in chat for in-line equations

Jasch1

so what do I change

Axoren

Post here what you have

Jasch1

"$\displaystyle\lim_{c\to\infty}\dfrac{f(c)}{c}$"

not in quotes tho

$\displaystyle\lim_{c\to\infty}\dfrac{f(c)}{c}$

Axoren

08:13

Right, how does it show up in ShareLatex?

Steve

@Alessandro No

Axoren

You can use the "upload..." function in chat for pictures.

Brody

Does the default ShareLatex template have the appropriate packages integrated?

Jasch1

where is that?

Axoren

@Brody It might need \usepackage{amsmath} and \usepackage{amssymb}

Jasch1

08:15

the fraction isnt showing up

Alessandro

ok, this is in fact an isomorphism of vectors spaces, so those $2$ matrices on the left are independent in $\mathbb{R}^{2\times 2}$ iff those $2$ vectors on the right are independent in $\mathbb{R}^4$

Axoren

@Jasch1 Add these two lines to the beginning of the file, after documentclass

Brody

@Axoren Alright, that might be the issue. But you know more than me regardless

Axoren

\usepackage{amsmath}
\usepackage{amssymb}

Jasch1

that worked

Axoren

08:16

@Brody Wanna learn something neat? I don't think you were around for this silly answer. I still tell people about it because I like bragging about ridiculous things: math.stackexchange.com/questions/1817522/…

I just re-found this earlier.

Jasch1

@Axoren how could I easily explain limit

Axoren

In laymen's terms, it's the closest number that a sequence approaches if you were to do something forever.

Steve

@Alessandro Oh, I see. Nvm, there is a difference

Jasch1

But its a sum right

Alessandro

what do you mean

Brody

08:17

@Axoren Is it $1,-1,-1$ repeated?

Jasch1

@Axoren is it a sum tho

or not

Axoren

@Jasch1 A limit is not a sum. The limit of $2, 2, 2, ...$ is $2$

Hey-men-whatsup

value of function approaches some values of domainns?

Jasch1

yea but we were adding together all of them

Alessandro

@Steve have you seen that vector spaces are determined, up to isomoprhism, by their field and dimension?

Jasch1

08:18

nah i get what limit is

Axoren

@Brody Close, it's $1, -1, -1$ repeated.

So, it's a 3-periodic sequence.

Brody

@Axoren i dun ged it

Jasch1

Wait so for getting the natural density, is it just the limit?

Axoren

I managed to relate a very simple high school-level problem to Euler's formula.

Brody

I'm also hungry at +3 am.

Vrouvrou

08:20

hello, i need help for this

Axoren

Let's bake cookies together, @Brody

Vrouvrou

0

Q: Convergence in Orlicz Spaces

I have that $(u_n)\subset W^{1,\Phi}(\Omega)$ such that $u_n\rightarrow u$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous . How to prove that $$\int_{\Omega} |f(u_n)-f(u)| |v| dx \rightarrow 0, ~\text{for all}~ v\in W^{1,\Phi}$$ thank you

Jasch1

@Axoren before you were describing it as the sum of ratios

Axoren

@Jasch1 Yeah, the natural density is a limit of all those ratios.

@Jasch1 Not the sum, it was the limit. I don't think I ever said sum.

Alessandro

that's a very complicated way to say that the fibonacci numbers are twice odd, then once even, then twice odd, then once even and so on

Brody

08:21

@Axoren Too long. Me hungry

Jasch1

Are you sure about this?

Steve

@Alessandro Thanks for the explanation, but I need to sleep on it. I think I got the main idea of what you're saying though

Axoren

@Jasch1 If it were a sum, after 2 summands, you'd have over 100% density of any subset that contains both $0$ and $1$

Alessandro

no problem, I have to run too now, just ping me if you still have problems with this, I'm often here

Jasch1

true

Brody

08:22

Oh @Axoren, I see the formula in your comment. Lol applauds

Axoren

Because you'd have $\frac 1 1$ and $\frac 2 2$

Steve

@Alessandro Alright :)

Jasch1

I missread something from before

Brody

test: $\huge\text{Hello world.}$

2

Axoren

@Jasch1 Until you learn techniques for computing limits, you're going to have to imagine what the "last" element of an infinite sequence is.

The limit is the value which is the closest thing to the "last element" if one would exist.

Brody

08:24

:33863134 $\Huge\text{Huh?}$

Jasch1

i undertsand what it is, i just misread an old post, i though some commas were plus signs

Null

@Brody is this the giant language?

Brody

@Null No, dwarven.

They're quite loud and rambunctious.

Null

@Brody hahaha

Brody

It's compensation

Your formulation is going to bother me all night @Axoren

Axoren

08:26

@Brody I got distracted while writing "overcompensating?"

@Brody Which?

$a_n = (-1)^{F_n} = e^{i\pi F_n} = \cos(\pi F_n) + i\sin(\pi F_n)$?

Jasch1

@Axoren Can you explain why f(c) < c

Brody

13 mins ago, by Jasch1

$\displaystyle\lim_{c\to\infty}\dfrac{f(c)}{c}$

Axoren

@Jasch1 $c$ is how many you've seen, $f(c)$ how many of those $c$ that you've seen that were in your subset.

Brody

Bother me... at least for the few minutes I remain awake until keeling over

Axoren

You can't have found more numbers than you've seen

Jasch1

08:29

Can you explain the formula to me

I get the limit

But what does c respresent

Null

i have to go to the doctor but i don't feel like it. unsovlable problem

Axoren

@Jasch1 How many of the naturals you've checked so far.

Taking the limit means that you've reached the point at which you've checked them all

Jasch1

and func(c) represents?

Axoren

@Jasch1 First let me ask you a question. You remember what you're calculating the density of, right?

Jasch1

Yes

An infinite sequence

Axoren

08:32

@Jasch1 Not exactly.

Jasch1

which is the same size as the set of natural numbers

Axoren

@Jasch1 Natural density is defined on subsets of the natural numbers.

Jasch1

yup

Axoren

While infinite sequences of natural numbers can be seen as sets of natural numbers, this also includes finite sets.

Jasch1

which are of the same size (if infinite)

correct?

Axoren

08:34

That's a matter of definition and isn't really relevant.

Jasch1

ok

Axoren

It is true that if they're infinite they're the same cardinality as the naturals

Jasch1

So lets say we are checking somethign that returns something greater than c

Axoren

$f(c)$ is part of an iterative process.

You're checking numbers from your subset one number at a time.

$f(c)$ is how many numbers you've found after checking $c$ numbers.

Jasch1

$\frac 1 1,\frac 1 2, \frac 2 3, \frac 2 4, \frac 3 5, \dots$

so this

Axoren

08:35

Those are the values of $\frac{f(c)}{c}$

Jasch1

nah they were something/2

Axoren

Specifically, those are the ratios for the set of even numbers.

Jasch1

yes

Axoren

"nah they were something/2"?

Jasch1

sorry lmao

Axoren

08:37

In that case, $f(c) = \frac c 2 + 1$

That may be what you're confusing.

You don't actually need a function expression for $f(c)$

It just represents a quantity related to the process

Having one allows you to calculate the limit easily.

Brody

At least for the primes, we've actually denoted a counting function

Axoren

@Brody With a closed form?

Brody

$d(\mathbb{P})=\lim_{n\to\infty}\dfrac{\pi(n)}{n}$

No, just a symbol

Axoren

Meh.

Jasch1

f(c) can be replaced with a linear expression correct?

Axoren

08:39

If I remember correctly, that density is non-zero.

DHMO

@Axoren it is zero

Jasch1

Ok

Axoren

@DHMO That's a shame. I might be thinking of a proof that uses partial densities.

Brody

Well, any counting function on natural numbers has a closed-form expression, depending on the definition

Axoren

@Jasch1 It's not so much that $f(c)$ can be replaced by linear expressions

Jasch1

08:41

modified linear expressions

Axoren

And more that the counting functions of some nice sets happen to be linear

Jasch1

$\displaystyle\lim_{c\to\infty}\dfrac{2c}{c}$

Axoren

@Jasch1 You need to be careful with doing things like just picking an equation for $f(c)$

You get $f(c)$ from a description of a subset, not the other way around.

Jasch1

it would need to be m/2c right?

lets say we want to get all even numbers

Axoren

@Jasch1 Constructing $f(c)$ for $2\mathbb N$ would result in $f(c) = \frac c 2 + 1$

Jasch1

08:43

$\displaystyle\lim_{c\to\infty}\dfrac{\dfrac{m}{2c}}{c}$

You're confusing me a lot here

Axoren

$f(c)$ is a counting function of a set. If you check $c$ natural numbers, you'll have found $f(c)$ of them in your set.

Jasch1

Lets say I have the linear expression 2x + 1 and I want to find the density of the generated sequence

What would the equation be then

you could drop the +1

and thenwhat else would you do

Axoren

So you're trying to compute $f(c)$ from a sequence $a_n = mn+b$, right?

Jasch1

yes

Axoren

As I was telling Brody, earlier, the inverse works sometimes.

Jasch1

08:47

Generally that works

Axoren

Which is to do the following:

$f(c) = \frac{\frac c m - \frac b m}{c}$

Roughly.

However, the limit is equivalent if we drop the $\frac b m$ term

Jasch1

ok

Axoren

If the fractions are round, things don't work

Brody

This transcript is gonna be looong

Axoren

@Brody I can't explain why $f(c)/c$ is equivalent to $a(n)/n$ tonight

Jasch1

08:51

Can you help me out and write something in LaTeX

Axoren

@Brody Do you still have gripe with it?
@Jasch1 What specifically?

Brody

@Axoren Their respective limits you mean?

Axoren

@Brody Of course

Jasch1

Set builder notation

Axoren

@Jasch1 $\{ x \mid P(x) = 1 \}$

Brody

08:52

@Axoren Is it just the case that your inverse function approximates the counting function at large numbers?

Axoren

Use \$\<space>\$ for spacing things out a bit

@Brody It has to do with the fact that the sets of counted numbers at each step are subsets of each other.

Brody

@Axoren Okay.

Axoren

If $A(n)$ are the sets counted in order, $B(c)$ are the sets counted in an order you choose.

$B(0) \subset B(1) \subset \dots \subset B(n) \subset \dots $

Jasch1

is there a sign for natural density?

Axoren

@Jasch1 A nice letter to use is $d$

That's what they use on the wiki

@Brody So, now consider the fact that that sequence of $B(c)$ converges to $\mathbb N$

You have a convergent sequence of sets of natural numbers.

Brody

08:58

What do you mean by an "order you choose"?

Axoren

@Brody I want to check if all the evens are in the set first, then if all the odds.

If that's the order you pick, it's not possible.

Brody

@Jasch1 Do you mean notation or positive/negative?

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$Mathematics$ Mathematics