Simply Beautiful Art's realm of calculus and analysis

Nilknarf

00:36

@SimplyBeautifulArt Hey!

I am learning the residue theorem! :D

Leaky Nun

nice

Nilknarf

:D

It is so much fun, I've wanted to learn it for so long

Leaky Nun

@Nilknarf in other news

27

Q: Is there a computable model of ZFC?

Background Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being the set of natural numbers, so $\in$ will be some binary relation on the natural numbers. Can...

sad story

Nilknarf

:(

Simply Beautiful Art

14:57

@Nilknarf NIIIIICE

@LeakyNun :o

Anant Saxena

16:06

Hey @SimplyBeautifulArt!

Leaky Nun

16:52

@SimplyBeautifulArt ?

Typhon

18:57

@SimplyBeautifulArt mind if I ask why you voted to delete an answered question?

math.stackexchange.com/questions/2499002/…

i fully get that closures arent that big of a deal especially if a post has been answered, but why delete..?

Simply Beautiful Art

20:50

@Typhon Because it appears as though you were talking nonsense to yourself over something you trivially answered, and there doesn't seem much to keep from that question. I see no reason to keep that question on this site.

Anant Saxena

Hey @SimplyBeautifulArt!

Simply Beautiful Art

@AnantSaxena hey

Anant Saxena

I came up with an interesting conjecture indicating a relation about a series

wanna see?

1

Q: Variant of Gilbreath's conjecture implications for a series?

Background Similar to Gilbreath's conjecture I tried to create my own conjecture. We start with the primes: $$ 2,3,5,7,11,13 $$ We then take the absolute difference between the terms side by side and call this the "same row difference step": $$ 2,3,5,7,11,13,\dots $$ $$ 1,2,2,4,4,\dots$$ No...

linear-algebra sequences-and-series number-theory conjectures

I'm super hyped about what this could mean if the conjecture is true :D

Simply Beautiful Art

oh dear lol

Anant Saxena

what?

Simply Beautiful Art

20:58

Perhaps we should get a program to compute this?

Anant Saxena

yea ... If i only had computer skills

(btw u read that pretty quickly I must say)

Typhon

@SimplyBeautifulArt i was not talking nonsense. I could easily say the same about your big number quests and analysis. Just because you dont understand something doesnt make it nonsense.

Simply Beautiful Art

@Typhon ._. Accusing you that your stuff is nonsensical is not the same as saying I didn't understand any of it. Please don't try to interpret what I've said in any way other than what I've said.

Typhon

fair enough

what makes you say it is nonsense then?

:/

Leaky Nun

@SimplyBeautifulArt hi :D

7

Q: BigNum Bakeoff Reboot

Some of you may be familiar with the BigNum Bakeoff, which ended up quite interestingly. The goal can more or less be summarized as writing a C program who's output would be the largest, under some constraints and theoretical conditions e.g. a computer that could run the program. In the same spi...

math code-challenge busy-beaver

why am I #4?

Typhon

21:13

I showed the question is virtually unanswerable as under that field extension multiplication is multi-valued and therefore my proposition in the question lacks any sensical meaning within the extended set

Simply Beautiful Art

@LeakyNun hm?

Leaky Nun

@SimplyBeautifulArt why am I not #3?

Typhon

it no longer makes sensw to talk about linearly independent elements that form "axes" in that number sets space as there is no such thing as linear independence

in that set

@SimplyBeautifulArt am i missing something here? cause i only voted to close as undecidable problems are not valid in mathematics

not because of clarity

Simply Beautiful Art

And?

> Because it appears as though you were talking nonsense to yourself over something you trivially answered, and there doesn't seem much to keep from that question. I see no reason to keep that question on this site.

Other than the subjective use of nonsense, what are you objecting to?

Typhon

youve never explained what you mean by nonsense?

Simply Beautiful Art

21:20

@Typhon And it's hard for you to argue on that point when my questions are generally +5 upvotes, there's an entire wikia dedicated to it, to which there is a serious mathematical community, etc.

> Other than the subjective use of nonsense, what are you objecting to?

@LeakyNun and idk

Leaky Nun

@SimplyBeautifulArt I thought you put me there

Simply Beautiful Art

Sh

Is fixed

Leaky Nun

eh

@Typhon just doesn't know how to construct things formally

which is actually his fault also, but I think it can be fixed

it's a quotient of all the real functions

Typhon

i do know how to construct things formally

Leaky Nun

under the relation $f \sim g \iff \displaystyle \lim_{x\to\infty} f(x)-g(x) = 0$

Typhon

21:24

@LeakyNun please do not make rude assumptions

Leaky Nun

@Typhon oh, then it's entirely your fault and I do think that your question is unclear =P

@Typhon I was trying to help

Simply Beautiful Art

@AnantSaxena =( How to use Ruby's prime numbers

Leaky Nun

@Typhon because if you knew, you would have known that $\displaystyle \lim_{x\to\infty}\sin(x)$ is not a number

Typhon

@LeakyNun REAL number

Anant Saxena

@SimplyBeautifulArt are you referring to this? teamtreehouse.com/community/finding-prime-factors-in-ruby

Leaky Nun

21:26

I wonder if you even know the definition of limit to begin with

Simply Beautiful Art

@AnantSaxena No

Anant Saxena

then?

Simply Beautiful Art

ur supposed to do a think like Prime[5] to get the 5th prime or something

Typhon

you're being needlesly rude and rejecting my question altogether

Simply Beautiful Art

But idk how

So I'm just gonna brute force my own primes

Typhon

21:27

the whole point of the question is to examine a number set where divergent limits exists

Simply Beautiful Art

What's a good upper bound to the nth prime?

Leaky Nun

@Typhon you're just avoiding my question

Simply Beautiful Art

sighs

Typhon

there was no question

there was a rude remark

Leaky Nun

Do you know the definition of limit?

Anant Saxena

21:29

not sure ...

Typhon

yes

Leaky Nun

what is it?

Typhon

ive seen the logic defining it

Anant Saxena

@SimplyBeautifulArt not sure

Simply Beautiful Art

@Typhon I think Leaky's point is that the limit you used in your question hasn't been defined yet, and that the normal definitions assume that expressions such as $\lim_{x\to\infty}\sin(x)$ do not exist.

Typhon

21:30

there is only definition of the limit

just as there is only one definition of square root

Leaky Nun

right, so define it

Typhon

yet we talk about complex numbers which dont exist

Leaky Nun

they do exist

they are formally constructed

Simply Beautiful Art

^

Leaky Nun

you're still avoiding my question

Typhon

21:32

they are the set of all rea number linear combinations of i and 1

Leaky Nun

yes

Typhon

i doesnt exist. its not a real number. by your own logic it doesnt exist

because it violates the definition of square root

Leaky Nun

sigh

Typhon

and how would i formally define a number set that im asking a question about for the sole purpose of characterizing

Leaky Nun

@Typhon I just defined it for you

Typhon

21:34

not all diveegent limits fit that

Leaky Nun

they do

Anant Saxena

@SimplyBeautifulArt Im trying to do it on Msexcel witht the first 50 primes

Leaky Nun

hmm, they don't

Typhon

"quotient of all real functions"

Simply Beautiful Art

lol

Leaky Nun

21:34

intuitively you want $\sin x$ and $\cos x$ to have the same limit

so the equivalence relation might need to be tuned

Typhon

precisely

Leaky Nun

so you still need to define it

and it isn't my job to define it

it's your question that you want undeleted

Typhon

not all things in math require symbolic definitions

Leaky Nun

then your question is unclear

things have precise definitions in maths

$i$ has a precise definition

and normal questions about complex numbers don't need a definition because we all know their precise definitions, at least we assume that we do

whereas your question presented novel concepts, which must be defined beforehand

Anant Saxena

@SimplyBeautifulArt apparently excel can b very frustrating to work with

Simply Beautiful Art

21:40

@AnantSaxena u just now realized

Anant Saxena

@SimplyBeautifulArt are you still trying to check it computationally or is my hopeless Ms-excel the last hope?

Simply Beautiful Art

I'm still writing a function that spits out primes

I like to do this stuff without using other people's codes

Anant Saxena

ohk ... because I feel like giving up on excel ... (time to learn Ruby)

Simply Beautiful Art

lol

tio.run/…

@AnantSaxena

Faster run time

tio.run/…

Terribly slow

Anant Saxena

21:57

yay! progress?

is beeing made??

Typhon

@LeakyNun wait. why are we trying to define what it means for two elements to be equal? the only thing we need to define is what is in the set.

or am i mistaken?

what makes two elements equal should be done as a theorem not a definition

or at least the exact properties beyond stating the limits to be equal

Simply Beautiful Art

@AnantSaxena Wait, how are you doing your 'differences'?

Typhon

@LeakyNun im assuming literature on the matter already exists?

Leaky Nun

@Typhon well think about how Cauchy defined the reals

Typhon

certainly theyve been already brought up considering we've had 400 years

Simply Beautiful Art

22:03

Oh

Leaky Nun

he defined it as a quotient of all the rational sequences

Simply Beautiful Art

Same row, then double row

Leaky Nun

@Typhon well not that I'm aware of it

Typhon

@LeakyNun i dont know how he defined them

Leaky Nun

if you use Dedekind cuts you won't need quotients though

@Typhon https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_fro‌m_Cauchy_sequences

Typhon

22:04

you are aware that my question comes from a computation need right?

i edited it out as i assumed that was the clarity issue

Simply Beautiful Art

tio.run/…

Typhon

my highest dipping of analysis is undergrad diff eq

Simply Beautiful Art

@AnantSaxena Verified up a few terms

Typhon

i know the limit def from calculus

not analyis

(though i will admit i dont have it memorized)

Anant Saxena

@SimplyBeautifulArt im taking absolute difference

also how many is a few?

Simply Beautiful Art

22:06

@AnantSaxena Yeah, that's what the .abs does

Be patient, I'm computing

Typhon

@LeakyNun surely the concept is clear enough though that people can state things about it, right

once you state all divergent limits to be convergent in such a set, then you immediately have that all limit properties hold for divergent limits that once only held for convergent limits

Leaky Nun

@Typhon it isn't really precise enough

Simply Beautiful Art

@AnantSaxena If the primes are the first row, then their differences are the second row, etc.

Leaky Nun

e.g. does $\sin(2x)$ and $\sin x$ have the same limit?

Simply Beautiful Art

Then I've verified up to the 14th row

Typhon

22:08

from that you can easily prove theorems stating what values are equivalent

Leaky Nun

aha, I think I know what you want

you want to quotient by $(\limsup,\liminf)$

Anant Saxena

That counts to be as progress

Leaky Nun

$f \sim g \iff \displaystyle \limsup_{x\to\infty} f(x) = \limsup_{x\to\infty} g(x) \land \liminf_{x\to\infty} f(x) = \liminf_{x\to\infty} g(x)$

Anant Saxena

*me ... thanks alot simply

Leaky Nun

@Typhon how can you prove something that isn't well-defined to begin with?

Typhon

22:10

@LeakyNun if we state all limits to be convergent then we can prove by limit properties which limits are equal

Simply Beautiful Art

Np @AnantSaxena

Leaky Nun

you can't just state all limits to be convergent

Simply Beautiful Art

tio.run/##PU7LDoIwELz7Hya7Ste23oTKF/…

You can run this program

Until it breaks at 60 seconds

And it returns what it has figured out so far

Typhon

@LeakyNun i can state there exists a number set where they are

Leaky Nun

you can't just do that

Typhon

22:11

and use that as the definition of the number set

why?

Leaky Nun

I'm typing

Typhon

how is that any different than using square root of negative 1?

to define complex

Leaky Nun

@Typhon that's imprecise also

the formal construction doesn't use sqrt(-1) at all

Typhon

O.O

i thought it used it as an axiom or as the definition

and appealed to the real number definition for the definition of real number linear combination

i thought all else was built by proof

Simply Beautiful Art

I DECLARE

Typhon

22:13

?

Simply Beautiful Art

That there exists a solution to |x|=-1

Typhon

yeah

and its not a real number

Leaky Nun

well you can define $\Bbb C$ as an algebra of $\Bbb R^2$ or just $\Bbb R[X]/\langle X^2+1 \rangle$

@SimplyBeautifulArt that's nonsense

Simply Beautiful Art

Well what is it

@LeakyNun that's the point

Typhon

God knows what that value is though

not me

Leaky Nun

22:14

@SimplyBeautifulArt LOL, well played

Simply Beautiful Art

@Typhon The point is that this actually can't exist

Typhon

i would argue it isnt nonsense

Simply Beautiful Art

For like normal math

Lemme go find that reference real quick...

Typhon

real numbers are held under the properties of real numbers

Anant Saxena

@SimplyBeautifulArt why?

Typhon

22:15

arguing a set exists with certain properties that isnt the real numbers an arguing it cant exist because of some real number property is in my eyes MORE nonsensical

Here's a good thing i read somewhere

"We never state something doesnt exist. We state that it doesnt fall within the set we are examining."

Anant Saxena

sorry for a moment I thought u were talking to me about |x|=-1 ??? whattt?

Typhon

if only i knew who wrote that

Leaky Nun

@Typhon when we said that the limit doesn't exist, we meant that it doesn't exist within the real numbers, so you need to define your new set and construct it formally

Typhon

@LeakyNun yeah im defining it as the set of numbers that fulfill the definition of the limit for all limits of all single variate functions as the input increases without bound

(with the exception of limits that increase without bound)

Leaky Nun

@Typhon and I'm saying you can't just state it like that

Typhon

22:18

why?

do all sets have to be written symbolically?

Leaky Nun

because that isn't a precise definition

Typhon

i never said it was perfect

Leaky Nun

it has nothing to do with symbolism

Typhon

but it is good enough to work with

Leaky Nun

symbols only help to make it more precise

I disagree

Simply Beautiful Art

22:19

278

Q: Why don't we define "imaginary" numbers for every "impossibility"?

Before the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had, so we said $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as $x = x + 1$, or $x = 1/0$? Edit: OK, a lot of people...

soft-question

0

Q: Is there a number whose absolute value is negative?

I've recently started to think about this, and I'm sure a couple of you out there have, too. In Algebra, we learned that $|x|\geq0$, no matter what number you plug in for $x$. For example: $$|-5|=5\geq0$$ We also learned that $x^2\geq0$. For example: $$(-5)^2=25\geq0$$ The exception for the $x^...

absolute-value

Leaky Nun

and so do the three people that voted to delete your question

Simply Beautiful Art

6

Q: Defining $|x|=-1$

Q 1a Is it possible to define a number $x$ such that $|x|=-1$, where $|\cdot|$ means absolute value, in the same manner that we define $i^2=-1$? I have no idea if it makes sense, but then again, $\sqrt{-1}$ used to not be a thing either. To be more explicit, I want as many properties to hold a...

definition recreational-mathematics absolute-value

Typhon

are we still arguing on whether or not the post should be deleted?

Simply Beautiful Art

There we go

Typhon

cause i never argued the question should be opened

Simply Beautiful Art

22:20

So no, you can't just say that some magical $x$ exists where $|x|=-1$

Typhon

i argued an answered question shouldnt be deleted

Leaky Nun

@SimplyBeautifulArt actually I have seen a natural norm which can be negative

it's the norm on $\Bbb Q(\sqrt2)$

Simply Beautiful Art

@Typhon That is a completely different question

@LeakyNun :P

Leaky Nun

$\|1+\sqrt2\| = (1+\sqrt2)(1-\sqrt2) = -1$

Simply Beautiful Art

35

Q: Under what circumstances is it appropriate to delete a question that has received a good answer?

I would like to have a discussion about the following question: When is it appropriate to delete a question which has been closed as "missing context or details," but which has generated mathematically good and upvoted answers? What considerations should be made when voting? The recent spat...

discussion voting deleted-questions deleted-answers

Leaky Nun

22:21

@Typhon $\displaystyle \lim_{x\to\infty} f(x) = L$ is defined as $\forall \varepsilon > 0 : \exists N : \forall x : x > N \implies |x - L| < \varepsilon$. I've been typing this for like minutes and I don't know where to start explaining.

so, forgive my poor explanatory power

Typhon

actually its defined as the L fitting that definition

Leaky Nun

eh, yes

Simply Beautiful Art

:|

Why is everyone using the bow as their hat

Leaky Nun

@SimplyBeautifulArt I earned a secret hat :P

Simply Beautiful Art

@LeakyNun which one?

Typhon

22:24

i see nothing stopping us from considering what if L wasnt a real number but some larger set

Leaky Nun

let's wait till it loads

"cosmic brain"

@Typhon because you haven't defined it

you can't just define $i$ to be the square root of $-1$

Typhon

eh

i would argue that depends on your level of formality

and if you can actually give a construction of such a number set go right ahead

im not stopping you

Leaky Nun

I'm saying that you need to construct it if you want people to take it seriously

Typhon

i initially argued that you said i was incapable of talking formally about math

@LeakyNun actually thats wrong

i need to provide a use

My primary reasoning behind this question is because one computational model for extending a thing called a turing machine is to construct a computational "machine" which can evaluate any limit.

Since divergent limits fall into that category it can produce such a value. Characterizing it as a space with a set of "axes" (similar to how we have $\{1,i\}$ forming the axes of complex numbers) might yield some manner by which we can construct a discrete version of such a machine and therefore make our turing machines more versatile.

^that is the use and my motivation

it doesnt hail from analysis

it hails from computation

Anant Saxena

about the sqareroot of -1 I've always wondered why must there be a unique imaginary number which satisfies it? For example why can't I have a four dimensional space where square root of -1 can be i, j ,k where i j and k are perpendicular to each other but this turns out to be simple vector algrebra which everyone knows about

Leaky Nun

22:27

actually I can construct your set in $\Bbb R^2$

Typhon

me?

Leaky Nun

@AnantSaxena well, for one, $(-i)^2 = -1$

and right, in the Quaternions you have $i^2 = j^2 = k^2 = -1$

@Typhon yes

Anant Saxena

yea ... but so is j^2 = k^2 =-1

Typhon

yer shitting me

Anant Saxena

yea ... thats what I was saying

Typhon

22:28

no way

prove it

Leaky Nun

lol

18 mins ago, by Leaky Nun

$f \sim g \iff \displaystyle \limsup_{x\to\infty} f(x) = \limsup_{x\to\infty} g(x) \land \liminf_{x\to\infty} f(x) = \liminf_{x\to\infty} g(x)$

Typhon

(i mean actually prove it cause im curious)

LOL

let me link something

Leaky Nun

just let $(x,y) \in \Bbb R^2$ be the set of functions $\{f : \Bbb R \to \Bbb R \mid \liminf f = x, \limsup f = y\}$

Typhon

math.stackexchange.com/questions/2482359/…

Leaky Nun

then $\sin$ and $\cos$ go to $(-1,1)$

Typhon

22:29

not all limits have a sup and inf

Leaky Nun

what do you mean?

Typhon

the linked question gives such an example in the answer

Leaky Nun

ah, there's a problem with my construction

you see, the equivalence classes aren't actually preserved under addition

(so it isn't actually a problem with my construction)

Typhon

wait

that example is floor

Leaky Nun

Take $f = \sin$, $g = \cos$, and $h = -\sin$

in your alleged set, they have limits $\overline f$, $\overline g$, and $\overline h$

and you said that $\overline f = \overline g$

now look at $\overline {f+h}$ and $\overline {g+h}$

Typhon

22:32

yeah

i know

i gotta go

Leaky Nun

see you

Typhon

link me if you have a proof

Leaky Nun

of what?

Typhon

yer statement that it is R^2

Leaky Nun

3 mins ago, by Leaky Nun

just let $(x,y) \in \Bbb R^2$ be the set of functions $\{f : \Bbb R \to \Bbb R \mid \liminf f = x, \limsup f = y\}$

didn't you see this

Typhon

22:33

i think that only works for nice functions

Leaky Nun

no, limsup and liminf exist for all bounded functions

Typhon

ok

i dont think all equal limits have equal limsup and liminf

Leaky Nun

what does that mean?

the (ordinary) limit exists for a function iff limsup = liminf

and the limit is the common value

i.e. $\lim f = L \iff \limsup f = \liminf f = L$

Anant Saxena

22:54

@SimplyBeautifulArt Big news!!! Someone verified the conjecture to first 100 rows!

Leaky Nun

@AnantSaxena what conjecture?

Anant Saxena

2

Q: Variant of Gilbreath's conjecture implications for a series?

Background Similar to Gilbreath's conjecture I tried to create my own conjecture. We start with the primes: $$ 2,3,5,7,11,13 $$ We then take the absolute difference between the terms side by side and call this the "same row difference step": $$ 2,3,5,7,11,13,\dots $$ $$ 1,2,2,4,2,\dots$$ No...

linear-algebra sequences-and-series number-theory conjectures

@LeakyNun this one^

Leaky Nun

sounds like a fun programming exercise

Anant Saxena

personally I didnt do it

simply verified the 1st 14 rows on ruby

and someone did the first 100 in the link

Leaky Nun

did your conjecture just get falsified?

@AnantSaxena

Anant Saxena

23:06

yea ... I think so ..

Leaky Nun

yay :D

Anant Saxena

y yay?!

Leaky Nun

because we gained knowledge

I think you can answer the question then

Anant Saxena

It seems so arbitrary that at line 104 it gets falsified

Leaky Nun

lol

Anant Saxena

23:08

What should I say the conjecture is not valid ... as in line 104 it does not work?

seems so random ... me so annoyed

>_<

Leaky Nun

@AnantSaxena yes

Anant Saxena

Maybe I should rephrase the question what goes right in the 1st 100 lines that it works?

I thought it was supercool that I had also a series that was a direct analog .. and now 48 pops outta nowhere(instead of 0)

Leaky Nun

N = 1000

# Eratosthenes sieve
primes = range(2,N+1)
i = 0
q = 2
while q*q <= N and i < len(primes):
	q = primes[i]
	primes = [p for p in primes if p%q>0 or p==q]
	i += 1

# rows
drd = primes
srd = [abs(primes[i]-primes[i+1]) for i in range(len(primes)-1)]

while srd:
	print(drd)
	print()
	print(srd)
	print()
	drd = [abs(drd[i]-srd[i]) for i in range(len(srd))]
	srd = [abs(drd[i]-drd[i+1]) for i in range(len(drd)-1)]

tio.run/##dY/…

my program

Anant Saxena

thanks for helping @LeakyNun ... You can post an answer too if u want ...

me gonna grumble about this strangeness for the rest of the day

Leaky Nun

@AnantSaxena twin primes

Anant Saxena

23:18

Ofcourse twin primes! how could I have forgotten about them

Leaky Nun

if you trace the diagonal from the 104th line

you will get to 239 241

if you trace the occurence of 10 in row 58

you will get to 197 199

Anant Saxena

So maybe the series is telling us something after all but fails for twin primes? Maybe my algebric representation still points towards a connection

Or am I being too optimistic?

@LeakyNun ?

Leaky Nun

eh, you haven't really took the absolute difference into account

it's an absolute difference, not just a difference

Anant Saxena

wait the programmer was wrong?

Leaky Nun

lol

Anant Saxena

23:25

ur saying he too difference instead of modulus difference?

Leaky Nun

no

Anant Saxena

ohk ... then what?

Leaky Nun

I mean, your algebra is wrong

Anant Saxena

I think u can take the absolute difference using my algebraic method ...

Leaky Nun

the data still look like pure luck to me

I mean, how the two big diagonals are generated

why they are exactly where they are

what determines the strength of the diagonal

looks nothing more than pure chance to me

Anant Saxena

23:28

As physicist in training I think we should just empirically check? Does this thing always fluctuate when there are twin primes?

fluctuate = go wrong at twin primes?

@LeakyNun the algebric method does allow us to study this behavior to some extent ... (I think)

Simply Beautiful Art

23:47

@AnantSaxena Cool!

Anant Saxena

@SimplyBeautifulArt which part of the failed saga is cool?

*failing saga

Simply Beautiful Art

Well still cool

Failure isn't the worst of things

It's what we learn from any experience that makes it memorable.

Anant Saxena

Im just wondering why does it mess up everytime (?) at twin primes ...

We found $2$ such daigonals which could be traced back to twin primes

Simply Beautiful Art

:P Prove it!

While I talk about large cardinals

hehe

Transcript for

$Mathematics$ Simply Beautiful Art's realm of calcu…