Mathematics - 2021-04-13 (page 1 of 3)

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12:24 AM

What can we say about the transformations: $(x,y)\mapsto (ax,\frac{y}{a^n})$ for $n \in \Bbb N$

for $n=1$ this is called a "squeeze mapping"

and for $n=1,$ $xy=\text{constant}$ is invariant under the transformation

so for $n=2,$ $x^2y=\text{constant}$ is invariant under the transformation

and for general $n$, $x^ny=\text{constant}$ is invariant

2

Q: About the transformation $(x,y)\mapsto (ax,\frac{y}{a^2})$

Suppose $f(x)=\frac{1}{x^2}$ and $g_k(x)=kx$ for parameter $k\in\Bbb R_{>0}.$ $g_k(x)$ maps onto itself as $k$ increases. The transformation maps a point on $f(x)$ to another point on $f(x).$ How would you write down this transformation more mathematically? Is it related to another, more well-kn...

linear-algebra linear-transformations transformation

related to this^

it seems like only $n=1$ has a name

maybe because it's the most important/useful

12:56 AM

what is a quick finite example of different groups with same subgroup lattices?

groups with intermediate subgroups

Oh nvm

woops

1:34 AM

It's kinda scary when you don't have ChatJax running, but you don't realize it and can read all the MathJax on the page.

that's when you know you've become a vampire

I don't have a bookmark toolbar so I never have chatjax running

I should learn how to configure a hotkey for it

on firefox

$f_n(x)=x^n$ are polynomials for natural number $n.$ What are $\frac{1}{f_n(x)}$ called?

I think it would be a Laurent polynomials consisting of purely negative $n$

StudySmarterNotHarder

1:54 AM

If $\text{Hom}(B, Q) \xrightarrow{f^*} \text{Hom}(A, Q) \to 0$ is exact and $Q$ is injective object (in an abelian category), then how does the proof go that $f : A \to B$ is a monomorphism?

@geocalc33 hey

@geocalc33 want to learn some homological algebra, which applies to all math in existence?

@StudySmarterNotHarder hey

StudySmarterNotHarder

hey, got banned for 2 days that why I disappeared suddenly

Yeah, I'm a badass

^_^

what did you say?

StudySmarterNotHarder

I dropped the i-bomb preceeded by the r-bomb

when people downvoted your post on fields

Hello everyone! Is this question clear? math.stackexchange.com/questions/4099959/…

StudySmarterNotHarder

1:57 AM

@newUser looks clear, but I don't know enough about Fourier analysis to answer

does anybody know if you can collect a bunch of linear maps and if it forms some algebraic structure in and of itself?

StudySmarterNotHarder

Yes, it definitely does

If those maps share codomain and domain

It forms a linear space, either an $R$-module or a vector space usually

what kinds of structures are admissible?

StudySmarterNotHarder

@geocalc33 not sure please clarify

@StudySmarterNotHarder you basically answered with this

StudySmarterNotHarder

2:01 AM

want to learn some homological algebra since you know how to quotient abelian groups from last time?

Quotient groups / modules become "measures" of how much something is or is not, it's neat

@StudySmarterNotHarder it forms a linear space, either R-module or a vector space

could it form a group?

StudySmarterNotHarder

Every vector space is a group

as well as every $R$-module

oh yeah

StudySmarterNotHarder

they have underlying abelian group structure by definition, and together with a scalar mult which makes them into $A$-modules for some ring or field $A$

When $A$ is a field you get by definition a vector space.

a vector space is a group with some extra structure

an abelian group

StudySmarterNotHarder

2:04 AM

Just that $r(sv) = (rs)v$ and $(r + s)v = rv + sv$ and $r(v + w) = rv + rw$

Where $R \times V \to V$ is the domain / range of scalar mult

It's like a ring acting on an abelian group, which if those three equations are satisfied, we automatically call it an $R$-module.

Each of those equations need to be satisfied, or the theory won't work

Call them the distributivity and scalar associativity laws

If $\phi : R \times V \to V$ is formally your scalar mult action, we write $rv := \phi(r, v)$

It's just notational to juxtapose

See how it works?

@geocalc33 yes, the space needs to be an abelian (commutative) group

Otherwise what you get is something else

@StudySmarterNotHarder I'm investigating whether this collection of linear maps forms a structure $(x,y)\mapsto (ax,\frac{y}{a^n})$

StudySmarterNotHarder

Where $n$ is fixed throughout?

$n=1$ is a specific linear map

StudySmarterNotHarder

Please explain more

each fixed $n$ is a linear map

And I'm taking the collection of all the maps

StudySmarterNotHarder

2:09 AM

$n$ is a natural number, or an integer, but we don't usually call those linear maps

$n$ is a natural number

why don't we?

StudySmarterNotHarder

You can say multiplication by $n$: $n \cdot : \Bbb{Z} \to \Bbb{Z}$ is a linear map

$n\cdot (x) = nx$

Because it is

the fact that n is fixed also makes your $(x,y) \mapsto$ map a linear map

StudySmarterNotHarder

But otherwise $n$ is just an element of the integers

but yeah in that context it's a number, parameter that you're using to define a family, and not itself a map (to me)

StudySmarterNotHarder

2:12 AM

Okay, so you're saying the space $V = \{ (x,y) \mapsto (ax, y/a^n) : a \in \Bbb{R}\}$ is a linear space?

A subspace of $L(\Bbb{R}^2, \Bbb{R}^2)$ the set of all linear maps?

for $n=1,2,3,...$ yeah

StudySmarterNotHarder

Is $n$ fixed throughout the problem?

Because if not, you might run into trouble maybe

If I fix $n$ there's only one map and hence no family of maps

i'm only seeing x and y in the rule definition but if these other things are varying in the analysis of interest i would definitely have more questions

StudySmarterNotHarder

There's an infinitude of maps, parameterized by $a$

Except $y/a^n + y/b^n = y/c^n$ probably only if $a$ is alowed to be in $\Bbb{R}$.

That would be required for $V$ above to be closed under $+$

But I'm not sure even then if closure happens

You'd definitely have to prove it

2:17 AM

So I can take the set of of linear maps $L=\{X_1,X_2,\cdot\cdot\cdot, X_n\}$

StudySmarterNotHarder

What are $X_i$?

$X_1=\{(x,y) \mapsto (ax, y/a^1): a \in \Bbb R\}$

StudySmarterNotHarder

I see

You'd have to prove that $y/a^n + y/b^m = y/c^k$ has a solution $k, c$ for all $a, n,m, y$.

First if $y = 0$ then any $c^k$ will do

So assume $y \neq 0$

Divide out $y$

Then mutliply by $a^n b^m c^k$

That takes things from denominators into numerators

You'd also need that simultaneously $a^n x + b^m x = c^k x$ for the precise same choice of $c,k$ as the first problem

So you have a system of two equations to solve, but because $c^k$ could be possibly many things, i.e. you're given two parameters to choose, to satisfy the two equations, you might have a linear space.

But in any case, $L = \bigcup_{i = 1}^{\infty} X_i$ not elementwise

like you have it

oh okay

maybe I could include things like $(x,y)\mapsto (ax,y/a^{1/n})$

StudySmarterNotHarder

Then $n$ becomes a rational

If the first one doesn't work, it would be worth a try

First, you should probably allow negative $n$'s in

Which would give the space more symmetry in definition

2:30 AM

okay

@StudySmarterNotHarder for $n=1$...Therefore, the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers

$n=1$ are called squeeze maps

StudySmarterNotHarder

nice

@geocalc33 :math.stackexchange.com/questions/4100099/…

Here's a post I made on my question

When an object of a category is injective, there are equivalent definitions, one of which I'm trying to prove

*finish the proof of

nice!

StudySmarterNotHarder

It's abstract non-sense for me at this point, but once about 5x more things are under my belt, I'm able to start working with Group Cohomology, etc.

Thx for the upvote !

@StudySmarterNotHarder if the collection of $n=1$ forms a one-parameter group, maybe the collection for every $n$ forms a one parameter group

StudySmarterNotHarder

True

2:36 AM

and then I'd be collecting a bunch of 1 parameter groups

StudySmarterNotHarder

Since you're dealing with rationals anyway, might as well take negative exponents

yeah

StudySmarterNotHarder

@TedShifrin hey :)

Hey :)

Andrew Micallef

2:50 AM

Quick question (will be back in a couple of hours for more details): if I have an equation of a second derivative of a function, can I integrate the other side of the equation to make the expression one of a first order derivative?

I want an expression for the first derivative with respect to x in the schrodinger equation

@TedShifrin Hi

andrew i am not an expert on the schrodinger equation but very frequently you can indeed integrate both sides of "f''(x) = blah" and get "f'(x) = new blah." the issue is that new blah might not be any easier to analyze than the original formula for f''(x). it's not guaranteed to be a slam dunk unless the RHS is in some way 'simple' or cleverly dealt with.

StudySmarterNotHarder

@AndrewMicallef that sounds like a good question for physics SE or MSE

I'm not a Schrodinger cat yet, so I couldn't tell you the answer

I did just purchase Griffiths though, paid $75

Still on the first chapter

It's astounding to me that particles obey these wave equations, what an amazing theory QM is

you see an analog of this in high school algebra, particularly when students move just one copy of something involving a variable x to the one side and then tries to solve for x by applying an inverse function to the other side. if the other side still has x's in it, that might not be much easier to analyze.

3:05 AM

@AndrewMicallef with regular derivatives, sure, but with partial derivatives you have to remember that the constant of integration is a function of the remaining variables.

Hi, Karim.

and that. often assisted by the not-always-helpful tendency in physics to omit things like bounds of integration or what depends on what.

StudySmarterNotHarder

@AndrewMicallef it seems to me that if you integrated w.r.t. either $x$ or $t$ (say the 1-D Schrodinger) you'd be left with an integral of $\Psi$ itself, not just an equation with $\Psi$ and its derivatives.

@TedShifrin I am thinking now of working out the surjectivity of that map by looking at surface theory. In particular Beauville's book on classification of surfaces. I think the surjectivity can be achieved by by dividing it into cases.

will discuss with my supervisor tomorrow. I don't think there exist one technique to prove it for all surfaces without delving it into cases.

Hi @leslietownes

Kodaira was such a boss.

good evening

StudySmarterNotHarder

@AndrewMicallef is $\dfrac{\partial \Psi}{\partial t} = \dfrac{i \bar{h}}{2m} \dfrac{\partial^2 \Psi}{\partial x^2} - \dfrac{i}{\bar{h}} V \Psi$ the equation you're dealing with?

See integrating with respect to $x$ say, and you'd be left with an integral $\int \Psi \rm{dx}$ term

3:11 AM

Yeah Kodaira so cool I will read his books definitely at some point.

StudySmarterNotHarder

If it doesn't work in the $1$-D case, it probably (QM is about probability anyway!) won't work in higher dims

@AndrewMicallef did that help at all?

3:26 AM

@StudySmarterNotHarder @Andrew Not to mention you have no idea what to do with the left-hand side. You can interchange integral and derivative and then you'll have $\frac{\partial}{\partial t} \int \Psi\,dx$.

BTW, it's $\hbar$, not $\bar h$ :P

StudySmarterNotHarder

Nice, $\hbar$ I was also writing incorrectly on paper

fixed

Right now going through the proof that a wave solution normalized at $t=0$ stays normalized as $t$ evolves. Learning some stuff :)

I haven't messed with calculus since high school essentially, and am new to $\Bbb{C}$ analysis, but the author knew this

Surprisingly the table of integrals at the back of Griffiths is only 9 items! Usually these tables have a hundred or so entries

@leslietownes. Hello. Hello All.

What is the general idea from Borel Sigma?

Is that a planet?

I read Wikipedia about Boreal Sigma algebra, but what is the idea behind them if you have read about them before?

(1) sigma algebras are cool, and (2) should you happen to have a topological space, why not try making one out of its open sets

3:33 AM

It is a general collection of sets on which we can consistently define a measure.

the Boreal sigma algebra is harsh and foreboding and somewhere in the artic circle

it involves trees in some strange way

StudySmarterNotHarder

Isn't the borel sigma algebra just the one generated by the topology?

*Borel

yes.

StudySmarterNotHarder

My memory is working :)

today at least -_-

3:35 AM

@copper.hat. @leslietownes. Thanks a lot. That is very helpful.

i always thought more in terms of integration than measure. integration behaves better if you can do a lot of countable limity stuff with functions. functions of interest to topology often behave nicely with the topology, so consider that when you try integrating them.

there's a bunch of stuff left out of that but that's my first few thoughts.

StudySmarterNotHarder

Read once or twice through the definition of Lebesgue measure then use basic properties thereafter is my philosophy

it depends on what you want to use it for. if you are a probabilist it is your bread & butter

there you sometimes are working with a completion, which can be worse for some purposes. books aren't consistent about this. but yeah, the standard examples are best.

StudySmarterNotHarder

The Lebesgue measure is the inf (or sup?) of an increasing family of simple functions

Or something like that :)

3:37 AM

noooooo

StudySmarterNotHarder

o__o

How do you find the T Statistics you have the standard error (1), Significance level (0.05), and sample mean (78.6)

probabilists often want integrate/average over sequences of random variables. that's another good test case, maybe more natural for some people.

the way i like to define it is the outer measure induced by the length of an interval (rectangle). in this i am heavily influenced by chernoff

StudySmarterNotHarder

Terrence Tao is a famous probablist, I can tell from his blog posts

3:39 AM

he is???

StudySmarterNotHarder

lol

certainly famous, not sure about the probabilist part

he's a famous lot of things. i think he is very comfortable with probabilistic notions, maybe more than a lot of analysts, geometers, etc.

im guessing you guys dont like stat

@StudySmarterNotHarder. Thanks. Usually, instructor starts with topology and borel before startign Leb measure. Are you suggesting the opposite?

3:39 AM

he does have a whole lot of expected values of stuff on there. or did, the last time i looked.

what is stat?

statistics

sry, i was just being a butthead

StudySmarterNotHarder

@Avra no, that's the right approach as outlined in Rudin

No problem

3:40 AM

@copper.hat Thanks Mr. Copper. So it's very important for statis, which is my case obviously!

StudySmarterNotHarder

Stat is when in the hospital they need something done right away

I had a question about finding the T-Statistics

StudySmarterNotHarder

What are those?

Can you define them for us?

@Avra just as the open sets define a standard topology on $\mathbb{R}^n$, the define a $\sigma$ field and an associated measure (Lebesgue).

@StudySmarterNotHarder. Doctor Rudin starts by leb measure before borel sigma?

StudySmarterNotHarder

3:42 AM

@Avra no, your approach is the best route

Test statistics. The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test. @StudySmarterNotHarder

StudySmarterNotHarder

Jesus, it would take me a week to learn up to that (at least)

Maybe @LeonhardEuler would know

@KyleReynolds what's wrong with the first definition given here:

In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is very similar to the Z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the t-statistic is used in estimating the population mean from a sampling distribution of sample means if the populati...

i'm not sure i would want to learn integration/measure theory from rudin?

@StudySmarterNotHarder. @copper.hat. Thanks a lot again

StudySmarterNotHarder

@KyleReynolds they give a formula which almost matches your parameter variables

3:45 AM

@Avra you are welcome, albeit i did nothing!

Folland is pretty good for measure theory

StudySmarterNotHarder

Since it is "defined" to be that, you can simply use the definition

Well I am not much interested in stats

StudySmarterNotHarder

You have to click the link to see the formulae

@LeonhardEuler prime numbers then?

$\Bbb{C}$ analysis is your bread and butter.

x-x^_/ S

StudySmarterNotHarder

3:47 AM

@LeonhardEuler are you / were you ever a victim of the prime numbers?

thats the one you're talking about @StudySmarterNotHarder

Were

StudySmarterNotHarder

wikimedia.org/api/rest_v1/media/math/render/svg/…

i only use subprime numbers. the primes are waaay too expensive.

are you now or have you ever been a member of the prime number party

StudySmarterNotHarder

3:48 AM

lol

I've been victimized by the primes. Even though they are a static structure

I am not always a math worm these days i am studying other things

they have awful trouble finding seating at restaurants

StudySmarterNotHarder

@LeonhardEuler what other subjects?

@StudySmarterNotHarder assembly programming, ethical hacking, etc.

ethical hacking????

StudySmarterNotHarder

3:50 AM

That's cool. I was a CS major one time

Yep

is the advanced class for unethical hacking?

StudySmarterNotHarder

I started programming by entering opcodes manually into a TI-86 calculator. You could then execute the text file as an assembler program using the OS, by default!

I had a table of all the opcodes

I had to translate them into hex numbers with address and everything. From there I moved into C

t value is 0, p vlaue is 0.5 and sinc ethe p is low i reject the h0

thanks a lot @StudySmarterNotHarder

StudySmarterNotHarder

@KyleReynolds you'll have to speak $\LaTeX$ here for people to read your math

Do you have the "Start ChatJax" link?

Google "math.stackexchange start chatjax" and bookmark to get formulas to display here for you

3:53 AM

no, I'm very new to stack exchange

StudySmarterNotHarder

Well, you surround your latex code in $'s

i had a hp-34c. still have my slide rule.

StudySmarterNotHarder

or $$'s for a large block of LaTeX

$$Since the p is low i rejected the H0$$

StudySmarterNotHarder

Right click any math you see posted, and "Show Math As.." > "TeX commands" to see how to write something, works on main MSE site too

Use \text{some text} if you want to put text in your formula

3:56 AM

Oh Okay

StudySmarterNotHarder

Otherwise use "some text $a^2$ some other text"

@StudySmarterNotHarder. Thanks for clarifying that to him.

https://www.math.ucla.edu/~robjohn/math/mathjax.html

StudySmarterNotHarder

@KyleReynolds Avra posted the link for you, drag the first Start ChatJax link into bookmark bar. You have to visit that link from your bookmarks each time you enter chat to render math

*first link on linked page

Open that link and add start ChatJax as bookmark. Then whenever you are here, please click on that bookmark for start ChatJax to render LaTex equations. This is how people ae able to read your questions Kyle.

Make sure after you got questions from teachers here to update your page by clicking on bookmark you have just saved for start ChatJa

*Make sure after you got answers :/

StudySmarterNotHarder

@Avra not sure what you mean on the last sentence, but mind you I'm high

on caffeine

:| I see now :>

3:59 AM

im in the post burger phase of the evening

@StudySmarterNotHarder. Haha, that is fine, but it's very late for Caffeine now

we're not all in the same timezone

StudySmarterNotHarder

Yeah, I'm a bad addict, I'm also smoking again, but it's my last pack - promise, for real

@copper.hat. My mistake :/

StudySmarterNotHarder

50% of the people are in my timezone because of UC Berkely et al

4:00 AM

at least i am assuming so! i on in pst

StudySmarterNotHarder

I'm in San Diego!

albany, ca

StudySmarterNotHarder

CA bros

:-) albeit i am from ireland

StudySmarterNotHarder

Albany is in NY, dude you don't know what state you're in :D

4:02 AM

it used to be ocean view

but the tired of getting mail for other ocean views

StudySmarterNotHarder

Nice, Irish bros too, my family is part Irish

so the mayor (from albany ny) renamed the city

StudySmarterNotHarder

That's nice, they can change our world like it's Python code. Very comforting !

el cerrito up theroad used to be known as rust

and berkeley is named after bishop berkeley of cloyne ireland.

even though he was never there.

and you thought this was about math...

StudySmarterNotHarder

I'm not picking up your Irish accent through text

I bet it's mind blowing

4:06 AM

i am guessing i have been in the usa for longer than you have been alive :-)

StudySmarterNotHarder

Oh, you lost your accent then?

I bet you can do a killer impression of an Irish lilt though

it appears in certain circumstances :-).

StudySmarterNotHarder

It's weird, people growing up around each other in the same country speaking the same language as other (English-speaking) countries, an accent develops, and it sticks over evolutionary time

*in that area

I'm going to try and hear your text in an Irish accent for bit

And I will detect from text-alone that you are truely from Ireland

you won't be able to understand what i am saying then

StudySmarterNotHarder

Yep, that came through Irish.

Always makes me wonder what the American accent sounds like to other areas

4:11 AM

there are four irish folks living in the same street here

tv means everyone has heard it

my hero jack lord

StudySmarterNotHarder

Tare are four Irish folks livin in tha saym strite hair.

That's what I'm hearing now

you might be a tad disappointed with the reality

StudySmarterNotHarder

LOL ^_^

a tad disapinted with the realitay

back to subprime numbers.

StudySmarterNotHarder

What are those?

4:14 AM

cause of our last financial disaster

jk

StudySmarterNotHarder

Semiprimes then?

pseudo has been taken already

StudySmarterNotHarder

Zhang has proven that there exists a $2k \lt $ 70 million about such that $X^2 - k^2 = pq$ infinitely often at least

maybe rational-assed primes.

StudySmarterNotHarder

$pq$ being a semiprime

4:16 AM

such as half-assed primes.

i have a feeling i am going to be struck by the math gods.

something-like primes

rib-free primes

not-primes

StudySmarterNotHarder

If you can prove that for instance there are infinitely many solutions $X$ to $X^{2^r} -1 = p_1 \cdots p_{r+1}$ then you've proved twin primes.

solutions $X \in \Bbb{Z}$.

and $p_i \in \Bbb{P}$.

too discrete for me.

4:19 AM

the twin prime conjecture is false. i have some wonderful pamphlets that lay out the whole thing. they are available for modest prices, given the importance of what is inside.

publishers don't seem interested in making a blind offer, so i turn to you, the public

for a premium the sale can be secret and you can claim the work as your own.

ahh, the shortsightedness of the mathematical public.

StudySmarterNotHarder

I think I'll take the last option :0

A real miracle of mathematics is that analysis is the strongest attack on prime number problems. In a sense Analytic NT is more powerful than Alg NT on prime number problems, all because of the zeta function and surrounding math

It's shocking that continuity and differentiation are used to attack discrete-like integer questions

look at fermat

StudySmarterNotHarder

That's not a strictly prime number related problem though

no, but has the same crossover

StudySmarterNotHarder

"Fermat's Last Theorem" it's like we're not taking any more statements from this guy since they take 300 pages of proof. Lol

4:27 AM

i think it should be possible to rebrand theorems...

generate some cash

StudySmarterNotHarder

Yeah, should be Fermat-Wiles-Shimura-Taniyama or something :)

McDonalds Inverse Function Theorem, sponsor of the 2021 MO

StudySmarterNotHarder

Actually add to that string 30 other names

Taco Bell Presents Fermat's Last Theorem

prime numbers with prime rib

4:29 AM

wiles needs a better sponsorship deals guy

StudySmarterNotHarder

Primality tests at the doctors office to see if you're a victim of prime numbers.

bend over

you've been factored.

The Civilised Glass

Love this thingy

that is sinful

StudySmarterNotHarder

Lol, they got the right kind of person to promote it too

Funny

4:33 AM

i hope no wine was hurt in the making of that youtube video

It is called the Pythagoras cup

His students were given this glass

www.kleinbottle.com/

oh goodness, cliff is still selling those

something about the bay area really attracts characters.

but caffe med is gone now so nowhere to congregate

they have to bother people at other cafes now. it was nice to have them all in one place.

well, bother people is too strong a word. congregate.

4:43 AM

there's gotta be a market for math sculptures. nothing as fancy as the bottle.

3d printers and whatnot

he must not like coffee, i never saw him at the nearest cafes. maybe he's only interested in things you can drink in a klein bottle.

a beer swimmmng pool

yeah it seems like there ought to be all sorts of things you could do fairly simply. with the right equipment. maybe that's the catch - too many peopple are doing useful things.

i once made an origami model of five intersecting tetrahedra, using some instructions on the internet. really tight tolerances. i would have preferred to print it out.

StudySmarterNotHarder

I drilled a 0.1mm hole into a piece of tungsten one time

I was trying to make a metal 3D printer

I have sense moved on and now obsess over math instead

@user863565 These lectures are base on book brown churchill complex variables. You can do exercises from there.

StudySmarterNotHarder

4:49 AM

@zacts any interesting NT questions you're working on today?

@AlessandroCodenotti what album is your avatar? I want to listen to that now :)

@JackOhara have you learned more Python in classes?

Mornin'

StudySmarterNotHarder

Evenin'

good night

Mornin'

StudySmarterNotHarder

What are your thoughts on perhaps the universe is a large fractal?

And we're actually in a static structure therefore

Time is an illusion

That explains why people seem to see fractals when they trip on halucinogens

at certain "stages"

It explains a phenomenon! It's a theory now :)

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