This is the Realm of Simply Beautiful Art

projectilemotion

4:00 PM

Hi

user21820

@projectilemotion: Hi.

Simply Beautiful Art

@projectilemotion What @user21820 said

projectilemotion

@user21820 How are you doing today?

Zophikel

@user21820 can I have a hint I have an idea on how to expand it

user21820

@projectilemotion: Doing fine!

projectilemotion

4:01 PM

@SimplyBeautifulArt And you?

Zophikel

it would be with the dealt epsilon definition

Simply Beautiful Art

@projectilemotion Normal?

Zophikel

But I don't think i'll get far without looking it up

Simply Beautiful Art

My thighs are sore

user21820

@Zophikel That's why you need to fully understand the definitions you come across, not just remember or attempt to use them.

2

Let me repeat what I said earlier:

projectilemotion

4:02 PM

@SimplyBeautifulArt I have a muscle ache

user21820

> Informally, we can say that for any positive error margin ε, there is a positive window of size δ around c such that if x is in that window then f(x) is close to f(c) within the error margin.

Zophikel

@user21820 I think I have it

user21820

Try it!

projectilemotion

@SimplyBeautifulArt I'm about to post a question

Simply Beautiful Art

@projectilemotion I'm about to try and read it

projectilemotion

4:04 PM

0

Q: A pattern in the general solution of the $n$-th order ODE: $\frac{d^n y}{dx^n}+\alpha x\frac{dy}{dx}+\beta y=0$ upon increasing the value of $n$.

Context: My friend (The same one who gave me this ODE) originally challenged me to obtain the general solution to this ODE: $$\frac{d^3 y}{dx^3}+\alpha x\frac{dy}{dx}+\beta y=0 \tag{1}$$ Where $\alpha,\beta \in \mathbb{R}$. I could not figure out any substitution without the use of power seri...

Zophikel

The definition can be extended as follows: $lim_{x->\delta} f(x) = f(c)$

Simply Beautiful Art

GAH

Beat me to it

projectilemotion

@SimplyBeautifulArt You read it fast lol

Simply Beautiful Art

lol

oh, its that thing

user21820

@Zophikel Um you're just repeating what you said earlier, which is wrong.

Zophikel

4:05 PM

@user21820 But I think what I did is wrong

@user21820 I know i'm not sure

user21820

You need to translate properly.

> for any positive error margin ε ( there is a positive window of size δ around c such that ( if x is in that window then f(x) is close to f(c) within the error margin ) ).

forall real ε>0 ( exists δ>0 ( ... ) ). Fill in the blank.

Zophikel

For all real ε > 0 there exists a δ>0 such that: $lim_{x->\c} f(x) = f(c)$

user21820

No. You're not actually attempting to translate what I wrote.

Zophikel

@user21820 oh

user21820

I'm asking to expand the definition of the limit, so you obviously can't use the "lim" notation anywhere otherwise you're just going in circles.

Zophikel

4:09 PM

@user21820 I know i'm just going in circles i'm stuck

user21820

And stop using English. It distracts from the logic.

Observe how I have translated some parts of my English sentence.

@user21820 <- Here.

Zophikel

@user21820 all right

user21820

And do the same for the rest.

Zophikel

@user21820 I think I have something close but it's still wrong

user21820

Fill in what you think is correct, and leave the rest blank.

Zophikel

4:12 PM

and I don't know how to latex quantifiers

projectilemotion

Stackexchange went down for a while

user21820

No LaTeX needed; I've not used any at all.

Just copy and paste what I wrote and replace the "...".....

You can use e,d instead of ε,δ for all I care. It makes no difference!

Zophikel

@user21820 I thought you wanted me to repharse the entire thing in quantifiers ?

?

user21820

Did you read what I wrote? I've already done it partially!

@user21820 <- Copy and add to this.

Zophikel

@user21820 I did i'm understanding what asking me to do i'm sorry ?

user21820

4:15 PM

Translate:

> for any positive error margin ε ( there is a positive window of size δ around c such that ( if x is in that window then f(x) is close to f(c) within the error margin ) ).

To logical form.

Namely:

> forall real ε>0 ( exists δ>0 ( ... ) ).

Zophikel

@user21820 ok I get it

But i'm having trouble with the last part " if x is in that window then f(x) is close to f(c) within the error margin "

Simply Beautiful Art

@TheGreatDuck Check this question

Hopefully you can help @projectilemotion

Zophikel

@user21820 i'm not sure on how to translate that

user21820

@Zophikel Well, the corresponding explanation for sequences is as folllows:

Simply Beautiful Art

@Zophikel Define "window" or "close to"

Zophikel

4:18 PM

@Simply ok

Simply Beautiful Art

You are given the error margin. I believe you can do this

Zophikel

Ok let me rewrite this for myself

user21820

I'll let @SimplyBeautifulArt continue for now.

Simply Beautiful Art

"close to" What determines how close something is to something else?

user21820

Partly because I have to go soon anyway.

Simply Beautiful Art

4:19 PM

@user21820 What? Aw man... I just had to but in...

user21820

@SimplyBeautifulArt: Thanks for the help!

Simply Beautiful Art

okay then, cya mate

Zophikel

" if x is in close to f(x) is close to f(c) within the error margin "

Simply Beautiful Art

lol

user21820

=)

Zophikel

4:19 PM

so the answer would like this

Simply Beautiful Art

No, if x is close to c within the error margin δ

Zophikel

@Simply ahh thanks

Simply Beautiful Art

then f(x) is close to f(c) within the error margin ε

Convert those two lines into stuff

Zophikel

If x -> C within $\delta$ then lim(f(x)) = f(c) within ε

@Simply I feel like there's a better answer

Simply Beautiful Art

Convert it into the logic

user21820

4:22 PM

@Zophikel: Let me just butt in for a few more minutes.

Tell me how to translate "one plus one is less than three" into symbols.

Zophikel

1+1 > 3

user21820

Wrong.

And then "five is close to four within an error margin of two".

@user21820 1+1 < 3

Closeness(5,4) < 2

@Simply all right

4:24 PM

Hahaha.. so @Zophikel, what is the symbolic form for "closeness"?

Zophikel

@user21820 i'm actually not sure

Simply Beautiful Art

What is this closeness function? Perhaps you should think something like metric, or distance between complex numbers or something

user21820

@Zophikel: Hmm so you don't understand the purpose of the absolute value function!

Simply Beautiful Art

But we only need the closeness function for two real numbers right now

user21820

"|x-y|" means what?

For real x,y.

Zophikel

4:25 PM

the distance form x minus y

user21820

No.

x minus y is a number. You can't take distance of a number.

You can take distance between two numbers.

Zophikel

@user21820 ahh all right I stated that wrong my mistake

Simply Beautiful Art

@Zophikel what does |x-y| mean for two complex numbers geometrically?

user21820

@SimplyBeautifulArt: Erm I'm not sure whether it's worth talking about complex numbers now.

Simply Beautiful Art

Well, maybe it's more intuitive to him.

user21820

4:28 PM

The point is that |x-y| is the positive difference between x,y, which you should understand as the distance between them.

Simply Beautiful Art

I mean, Idk, we'll see

user21820

On the real number line that is.

Zophikel

@user21820 I understand that

user21820

So @Zophikel you should know how to translate SBA's "closeness(5,4) < 2".

@SimplyBeautifulArt <- And hence you should be able to translate this too.

Zophikel

@user21820 now I have it

difference between 5, 4 < 2

^ positive difference between 5, 4 < 2 sorry

Simply Beautiful Art

4:29 PM

Not difference, distance

Zophikel

oh sorry

user21820

That's the whole point of using absolute value notation.

Simply Beautiful Art

Closeness, absolute value, norm.

user21820

Instead of saying distance(5,4) < 2.

We say...

what do we say @Zophikel?

Zophikel

@user21820 the norm of (5,4) < 2

user21820

4:30 PM

Symbols.

Simply Beautiful Art

Lol, no xD

Use absolute value bars and no words

|5-4| < 2

Okay, good

Yes.

@user21820 I'd like to help me with mathematical communication in the future

user21820

4:31 PM

So go back to what SBA wrote.

@SimplyBeautifulArt <- Here.

Translate it.

Simply Beautiful Art

x is close to c within the error margin δ

Zophikel

$x \rightarrow c$ For all $\delta$

user21820

You're not actually applying what we're telling you... =(

"five is close to four within an error margin of two" translates to?

x,c,δ are just names for real numbers just like five,four,two.

Zophikel

closeness (5,4) < 2 ?

user21820

Yes.

Zophikel

4:35 PM

@user21820 I know that

user21820

And we just told you not to use "closeness".

Zophikel

@user21820 i'm trying to find another way to express it

user21820

@Zophikel <- See what you wrote here yourself!

Zophikel

oh

user21820

Do the same with x,c,δ.

Zophikel

4:35 PM

|x-c| For all $\delta$

user21820

WRONG

That's not doing the same

You didn't write "|5-4| for all 2"....

So why are you doing that now?

Zophikel

@user21820 i'm having trouble with the delivery

I'm having trouble translating error margin

user21820

You must focus and listen. Not try to get to the result.

Zophikel

@user21820 ok

user21820

"five is close to four within an error margin of two" translates to?

"five is close to four within an error margin of δ" translates to?

Zophikel

4:39 PM

|5-4| < 2

user21820

"x is close to c within an error margin of δ" translates to?

Right.

Next two?

Zophikel

|x-c| < $\delta$

user21820

So what was so difficult just now??

It's correct!

Zophikel

@user21820 I just tired to figure it out and translate it myself

user21820

That is the real meaning of "|x-c|<δ"....

Zophikel

4:40 PM

I should have just asked for what I didn't understand i'm sorry :(

user21820

It means "distance between x and c is less than δ".

Zophikel

@user21820 I know I didn't do this without writing anything down

user21820

Now that's one part of the whole definition of the limit. I'll leave @SimplyBeautifulArt to continue as I really have to go.

But you're 3/4 way to finishing expanding the definition.

See it's not hard. Just systematically bit by bit.

So see you all!

Zophikel

@user21820 it's not that I don't have anyone to talk to rigorously about math in RL

user21820

Whether or not you do outside of Math SE, you've people here. =)

Zophikel

4:43 PM

@user21820 i'm trying to get to the point where I can be rigours without looking things up

It's my understanding that's the issues it's delivery of my understanding

user21820

You will get to that point. Just keep working at it.

See you!

Simply Beautiful Art

That's why you practice logic

@user21820 puns...

Er, references

Zophikel

@Simply I feel dumb

Simply Beautiful Art

@Zophikel okay, so you ready for next step?

Zophikel

@Simply yeah

Simply Beautiful Art

4:46 PM

Sh, font say that. You gotta put it into a rigorous definition :P

Zophikel

@Simply let's do this

Simply Beautiful Art

Okay, so now we want this:

all right

One moment

all right

4:50 PM

For all real ε>0, we want to show there exists some real δ>0 such that the when the distance from x to c is within an error margin δ, then the distance between f(x) and f(c) is within an error margin ε

Write for me "distance from x to c is within an error margin δ"

Then: "distance between f(x) and f(c) is within an error margin ε"

and then we'll work on gluing the pieces together

Zophikel

@Simply how would you express "within" ?

Simply Beautiful Art

Inequalities

Zophikel

|f(x) - f(c) | < ε

|x - c| < δ

Simply Beautiful Art

Okay, so we want it to work like this:

Zophikel

al lright

Simply Beautiful Art

4:53 PM

|x-c| < δ, then |f(x) - f(c)| < ε

Zophikel

ahhh all right so I was close

Should have put the than

@Simply what's next

Simply Beautiful Art

You would write it like this:
|x-c| < δ → |f(x) - f(c)| < ε

The little arrow is basically "the left makes the right"

Zophikel

ahh all right

Simply Beautiful Art

we want to show there exists some real δ>0 such that the when the distance from x to c is within an error margin δ, then the distance between f(x) and f(c) is within an error margin ε

Zophikel

There exists some real δ>0 such that: |x-c| < δ, then |f(x) - f(c)| < ε

Simply Beautiful Art

4:54 PM

Now we state that δ>0 and δ in R. I'm just gonna do δ>0

Zophikel

all right

Simply Beautiful Art

$\exists\delta>0(|x-c|<\delta \to |f(x)-f(c)| < \varepsilon)$

Got it so far?

Zophikel

Yes @Simply I can translate it back

Simply Beautiful Art

Can you add the last piece? For all real ε>0, we want to show there exists...

Zophikel

We went to show there exists the limit ?

Simply Beautiful Art

4:57 PM

Nope

We want to show that the distance between f(x) and f(c) can be as small as we want it

Zophikel

@Simply ahh ok

Simply Beautiful Art

The distance between f(x) and f(c) must be smaller than ε, and it must be smaller than ε for every ε>0

Zophikel

ok

Simply Beautiful Art

this way, ε can approach zero, hence f(x) will approach f(c), and thus we have defined continuity at a point

Zophikel

@Simply all right now I understand

Simply Beautiful Art

4:59 PM

Then can you write the full statement?

Zophikel

But I feel like I could improve on communicating without looking things up

Yeah

For all real ε>0, we want to show there exists some real δ>0 such that the when the distance from x to c is within an error margin δ, then the distance between f(x) and f(c) is within an error margin ε.The distance between f(x) and f(c) must be smaller than ε, and it must be smaller than ε for every ε>0

Simply Beautiful Art

No, I meant with symbols only

Zophikel

^ That's the English the last part in symbols would like

@Simply oh all right

Simply Beautiful Art

$\exists\delta>0(|x-c|<\delta \to |f(x)-f(c)| < \varepsilon)$

You have this part from me already

Think about what it says and what's missing for it to be finished

Zophikel

E is supposed to eplision

Simply Beautiful Art

5:02 PM

lol, epsilon*

Notice you don't have things in the right places

Zophikel

sorry :(

I know

Simply Beautiful Art

The "for every ε>0" should be first

Zophikel

oh

Simply Beautiful Art

it should be followed by "there exists δ>0"

etc.

Zophikel

Simply Beautiful Art

5:03 PM

Hm, what's the f(x)-F(c) supposed to be at the end? Isn't it already covered?

Zophikel

@Simply oh I think so my mistake

I was reading from the statement you typed earlier and attempting to translate it

@Simply but I think I got it now

@Simply how did you work on communication

Simply Beautiful Art

trial, error, intuition, MSE, reading

Zophikel

@Simply it's hard for me talk but when I read it and write it down it's perfect

Simply Beautiful Art

You'd be surprised what kinds of things you can write in symbols that you can't write in English

I have one such example

Consider the smallest number you can't write using n symbols in first order logic

You can write this statement in higher order logic, and though I said it in English, the statement "first order logic" is hard to grasp.

Nonetheless, this almost a valid statement, but boy, I have a hard time comprehending what it means

Zophikel

@Simply when I get a theorm definition etc I write it down break it down and glue it back together. For me understanding is if it can be stated in simple terms

Perhaps I should redo this but with rigor

Simply Beautiful Art

5:10 PM

perhaps yeah

you miss a lot of details if you never look at things truly rigorously

Zophikel

@Simply I think that's the issue that's been attacking me

Simply Beautiful Art

mhm, sounds natural

Zophikel

@Simply I go from Rigor -> simple detail

perhaps I should be doing is RIgor -> simple detail -> RIgor(in my own terms)

Simply Beautiful Art

Yeah, that should be better

but you need to be able to full loop back to the original rigor without looking at it

Zophikel

@Simply I think that's what I haven't been doing

Simply Beautiful Art

5:13 PM

You think that's what you've been doing, but are you sure? Looking at your own work makes you overly confident, which is why we have peer review

Zophikel

@Simply true I haven't been looping things back to RIgor(in my own terms) that's what I meant

Simply Beautiful Art

And for all people, there exists a small tweak in their mind such that the tweak is not found in another mind, which implies that two people will not be able to fully understand each other.

^ Put that into symbols :P

Zophikel

1

Q: Understanding $Lemma \, (1.6)$ to imply Lipschitz Continuity?

In the book "The theory of Ordinary Differential Equations" I'm having trouble understanding $Lemma(1.6)$ which assets Lipchitz Continuity defined below, specifically the gap in my understanding is within the operation taken within $(2.)$,specifically speaking the specifics of my question can be ...

differential-equations multivariable-calculus

@Simply let's go through this again ask me to explain Lemma (1.6) in full detail and rigor

Simply Beautiful Art

@projectilemotion any luck?

projectilemotion

@SimplyBeautifulArt Not yet xD

Zophikel

5:16 PM

@Simply ready

Simply Beautiful Art

@Zophikel I should learn what Lipschitz continuity is first though, right?

projectilemotion

@Zophikel It is a well written question, so I was actually the one who upvoted it previously (I was wondering why it didn't get much attention)

Zophikel

@Simply yeah I'm not looking anything up :D

Simply Beautiful Art

@Zophikel Well you've already looked at your book. I haven't done this stuff before

@Zophikel Have you worked with a Taylor series before?

Zophikel

@Simply yeah but I haven't done it in a long time

Simply Beautiful Art

5:19 PM

Do you remember how to prove a Taylor series converges to the function its supposed to represent?

Zophikel

I belive a Taylor Series is a infinite Series in the form of: $\sum_{n}^{\infty} \frac{f^{n}{n}

Simply Beautiful Art

n! on the bottom

Zophikel

AHh all right thanks @Simply , @Simply couldn't you do a Taylor approximation

Simply Beautiful Art

and I'd have to spend some time reading through Lipschitz continuity and how the differentials work and such

@Zophikel The distance between the function and the Taylor approximation ...

does not always converge to zero

Zophikel

@Simply all right

@Simply maybe you could ask @Projectilemotion

Simply Beautiful Art

5:21 PM

What do you mean?

Zophikel

@Simply for him to quiz me on the statement of the Lemma

projectilemotion

@Zophikel I don't know much about Lipschitz continuity.

Zophikel

@projectilemotion :( I want to practice being rigors I need someone to quiz me

Simply Beautiful Art

Ah... your definition of convexity...

I've been looking for a nice intuitive and rigorous definition of convexity for a while.

Ah shoot, I gtg

Well, I'll look through the stuff when I got time

Zophikel

See ya @Simply thanks for the help so it seems like the way i'm learning seems best

@Simply do you think it's my fundamentals that are weak or the delivery of my understanding

Zophikel

6:07 PM

@amWhy what do you thin k

TheGreatDuck

8:54 PM

@SimplyBeautifulArt that makes me think of certain equations that were studied by legendre way back when. they were of the form x^2y'' + xy' + y = 0

errr

with coefficients

Simply Beautiful Art

9:08 PM

@TheGreatDuck Well, hope you can help @projectilemotion

While I create Atlas PAIN

:D

amWhy

@SimplyBeautifulArt Stop being a PAIN in the butt! :P

Simply Beautiful Art

=P

TheGreatDuck

@SimplyBeautifulArt yeah, I will... since you asked so nicely.

Simply Beautiful Art

Did I?

TheGreatDuck

yeah

Simply Beautiful Art

9:17 PM

huh, okay then

@Zophikel math.stackexchange.com/q/2225367/272831

@Zophikel See the importance of writing the definition of the limit correctly. Can you spot what's wrong there?

Hehe

I wonder if I will reach an inaccessible cardinal today

amWhy

@SimplyBeautifulArt Well I came close in reaching a cardinal today; one was perching on the ledge immediately below one of my windows.

Simply Beautiful Art

Was it accessible?

amWhy

No, not accessible. I tried to access it, but it knew better and flew away...

Simply Beautiful Art

Haha, I feel the same way!

Zophikel

9:41 PM

@Simply yeah I see what's wrong with this

Guy switched two things around hold on

Simply Beautiful Art

what is it?

Zophikel

well the F(x)-F(a) < E should occur first

Simply Beautiful Art

:-) Yup

Zophikel

and the |x-a| < E should occur second

@Simply I feel like there's a more rigours way of say that

Simply Beautiful Art

Can you translate what the thing says in English?

Zophikel

9:43 PM

yeah but latex doesn't render for IE :(

Simply Beautiful Art

If the latex doesn't render, then you can see the code

which is basically most of the English

Trust me, logic is simple, you are probably trying to over think it

Zophikel

For all epislion greather than 0 such that the difference between x and a implies the absolute difference between F(a) and f(a) greather than delta

@Simply yeah i'm also reading through a basic real analysis book to fill my gaps

Simply Beautiful Art

less than delta at the end

Zophikel

@Simply thanks for pointing that out

@Simply how did I do

:)-

Simply Beautiful Art

Eh, okay to me

For me, it translates as follows:

No matter how close (or far) x is from a, the distance between f(x) and f(a) is a positive real value for any x,a in R

Zophikel

9:47 PM

@Simply that's better then mine

Simply Beautiful Art

which is a pretty useless statement

:-) Well, just wanted to share that with you

I'll be out for a bit and then back

Zophikel

@Simply yeah nice exercise

also I wonder why nobody has taken a look at my question

1

Q: Understanding $Lemma \, (1.6)$ to imply Lipschitz Continuity?

In the book "The theory of Ordinary Differential Equations" I'm having trouble understanding $Lemma(1.6)$ which assets Lipchitz Continuity defined below, specifically the gap in my understanding is within the operation taken within $(2.)$,specifically speaking the specifics of my question can be ...

differential-equations multivariable-calculus

TheGreatDuck

@SimplyBeautifulArt I'm stuck. I cannot proceed farther than
$e^{x}(\frac {\beta}{\alpha x})^x y = \frac{1}{\alpha} \int (-(e\beta)^{x}\alpha^{-x} x^{-x-1} y''') dx$

sorry man

i dont have time for this

i just hopped on to check my messages

:p

projectilemotion

@TheGreatDuck Hi, just wondering exactly what you mean by "Could we somehow use algebra to reduce the order?" I tried reducing the order and didn't succeed.

projectilemotion

10:42 PM

@SimplyBeautifulArt I got an answer!

2

A: An interesting pattern in the general solution of the $n$-th order ODE: $\frac{d^n y}{dx^n}+\alpha x\frac{dy}{dx}+\beta y=0$.

Define $s=-\frac{\alpha x^n}{n^{n-1}}$ and introduce Euler operators $\vartheta=s\frac{d}{ds}$ and $\delta=x\frac{d}{dx}=n\vartheta$. Lemma. $x^n\frac{d^n}{dx^n}=\delta(\delta-1)(\delta-2)\ldots (\delta-n+1)$. Corollary 1. $\frac{d^n}{dx^n}=n^n\vartheta\left(\vartheta-\frac1n\right)\left(\varth...