Just looking for any and all solutions of this functional equation...

- Kainui

Just looking for any and all solutions of this functional equation...

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- chestercat

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- Kainui

\[f^n(x)=f(x)^n\]
To help clear up the notation:
\[f^3(x)=f(f(f(x)))\]
\[f(x)^3 = f(x)*f(x)*f(x)\]

- dan815

1

- dan815

hmm im trying to think about matrix transformations

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## More answers

- dan815

is there a transformation where applying the matrix is like a unitary rotation

- dan815

and then vector can also be unitary so multiplying by itself will leave it unitary

- dan815

and there will be a solution wrt to some n

- dan815

|dw:1449634100983:dw|

- dan815

|dw:1449634653837:dw|

- dan815

yooo xD gimme some hints

- Kainui

Hmmm Well I dunno what you're doing haha but I did find several solutions and some of mine look like yours

- ganeshie8

.

- Kainui

Like I think you see the infinitely many complex constant unitary or w/e solutions you do, for example:
\[f(x)=i\] is a solution to \(f^5(x)=f(x)^5\), \(f^9(x)=f(x)^9\), etc...
I also found some other one that works too, by plugging in \[f(x)=x^a\] and solving for a in terms of n, but that's still particular special cases.
I would really like some single non constant f(x) that works for all n.

- dan815

yeah i know what u mean

- dan815

i think there is such a function

- dan815

basically like lets find somme unitary matrix solutiosn
then we will find eigen values of those matrices and get quadratics form of that
to generate our functiosn bruh1

- Kainui

I don't get that idea, like what do you mean?

- Kainui

like can you solve \(f^2(x)=f(x)^2\) as an example with matrices to show me

- dan815

okay like u know how all polynomial functions can be turned into matrix represented function

- dan815

like
for example
2x^2+xy+y^2=0
can be represented in matrix form

- Kainui

Ok yeah go on

- dan815

|dw:1449635364161:dw|

- dan815

now i was suggesnting we look for something in the eigen values of this matrix in the middle

- dan815

then we will have some diagonailzed form and stuff

- Kainui

sure but how does this relate?

- dan815

well okay like umm

- Kainui

I am familiar with all this sorta stuff you've said, another fun fact about symmetric matrices is unique eigenvalues of a symmetric matrix have orthogonal eigenvectors.

- Kainui

might be useful to us once we figure out how this connects to our problem I guess

- dan815

oh yaa i saw something like this before

- dan815

i forget the proof

- Kainui

It's pretty fun, not too complicate

- dan815

|dw:1449635780007:dw|

- dan815

|dw:1449635867198:dw|

- dan815

|dw:1449635981408:dw|

- dan815

|dw:1449636097991:dw|

- Kainui

let's say matrix A has these two unique eigenvalues a and b with eigenvectors x and y respectively. Show that \(x \cdot y = 0\) (they're orthogonal)
Ax=ax
Ay=by
Ok so you take these true statements, and we can left multiply by the other vector like this on both:
\[y^T Ax = a y^Tx\]\[x^T A y = b x^Ty\]
Transpose one of them, they're equal, so the rest follows cause \(a \ne b\)
\[\huge \square\]

- dan815

what do you mean transpose one of them

- dan815

|dw:1449636248477:dw|

- dan815

|dw:1449636293324:dw|

- Kainui

Yeah

- dan815

|dw:1449636363892:dw|

- dan815

|dw:1449636463431:dw|

- dan815

|dw:1449636480790:dw|

- dan815

i see this means that the only possibility of these 2 expression equaling each other would mean the dot product of x and y are orthogonal

- dan815

:)

- dan815

beautiful baby

- dan815

I AM A BEAUTIFUL BABY

- Kainui

http://static.guim.co.uk/sys-images/Guardian/Pix/pictures/2012/10/17/1350479348659/Crying-baby-013.jpg

- dan815

hahahaHAAHHAhahahaAHAHHAHA

- dan815

U R SO FUNNNY

- Kainui

lol calm down

- dan815

hshshshsahahah

- Kainui

YOU ARE OUTTA CONTROL!!!

- dan815

HAHAHa please delete all my msg that are compromising

- dan815

i am leaving you to it boss

- Kainui

Hahaha what

- Kainui

I'm not your post-janitor

- dan815

please editor!! think about it like editor

- dan815

You be the guy that decides on what comments of mine we should make public

- Kainui

clean up ur mess or I put u in time out. There's a reason moms = mods

- dan815

To bring up the image of your client

- dan815

YO NEXT question man next question!

- dan815

lemme catch up to your math knowledge a little
teach me some new stuff

- Kainui

lol idk what's to teach, tensors? I leave the country on saturday man

- dan815

LOLOLOl

- dan815

bruh pls plsplspls

- Kainui

I'm busy chillin doing w/e I feel like

- dan815

BRUH PLEASE!

- Kainui

I am doing some programming right now

- Kainui

and been doing some number theory nothing too serious

- dan815

okay lets do it! tell me what to make

- dan815

ill have it done for u rn

- dan815

okay whatever just gimme a prolem quickl man i need it bad i need it

- dan815

I NEED IT RIGHT NOW OMGGG

- Kainui

hmmm uhhh ok teach daggie

- dan815

LOOOOL

- dan815

LOLOLOLOLOLOLOLOLL

- Kainui

daggie is really smart and has a larger attention span than you, she'll easily pass you up

- dan815

does daggie get this eigen value stuff

- Kainui

she's reading this right now, not kidding didn't I show you that organic chemistry thing we did the other night?

- dan815

BRO how can u say that

- dan815

From now on iam back to math!

- Kainui

Do you get this eigen stuff? Teach her matrices

- dan815

BRO but i just got back after a 6 month break

- Kainui

hahaha ok

- dan815

NO i dont have time for such noobs

- dan815

loool!

- dan815

let the noob learn herself

- Kainui

I know but like I'll be gone for a month

- dan815

watch us and learn

- dan815

watch how the masters do it

- dan815

if its a question that interests me it will be solved

- dan815

simple as that

- Kainui

wowwwww

- dan815

:)

- Kainui

I'm reporting you to preetha ur no longer a qualified helper u hear me?

- dan815

OMG please dont stop okay im sorry dude

- Kainui

lolol xD

- dan815

LOL STOP IT

- Kainui

no more owl bucks how will u pay rent?

- dan815

LOOOL shush

- dan815

yo next question dude please!

- Kainui

Ok so dan you know how you said this stuff about

- Kainui

like "solving OS"

- Kainui

maybe we should solve PA instead

- dan815

um ya sure go on

- dan815

okay lets do it right now

- dan815

What do u want to solve?

- Kainui

since all they ask for is answers I mean we could just make like a PA bot

- dan815

okay how about we start with something simple

- dan815

like gathering data

- dan815

from a paragraph

- dan815

and fill out a form for us

- Kainui

idk, also my friend has this problem with code where he works with a bunch of drunk programmers or something that can't type well so he wants a program to parse drunk speech and if an instant message contains a certain garbled keyword or something, it will save that message.

- dan815

a form saying
Length of the question in words :
Number of times slope appears :
Number of times _____ appears:
....
Number of digits in text:
Number of groups of digits separated by spaces:
and so on:
Design it so that we have a easy way of implementing new tasks for it to do a find for, and also possibly a way of removing methods, or maybe just not using the methods for later on

- Kainui

ok dan how about we solve an unsolved problem instead like something simple like prove there are infinitely many twin primes, the collatz conjecture, goldbach conjecture, or riemann hypothesis

- dan815

yo bro i feel like i solved this right now!

- dan815

This is all we need yo

- dan815

It will be so sick listen to me man

- dan815

There will be 3 other things we we will keep track of for the seach methods

- Kainui

I hear you but the implementation of this code will be boring

- dan815

But dude we will not just be making a code here

- Kainui

it always starts out simple until you get into it and realize you're in a tangled web of garbage I've been down this path

- dan815

we are making a searching algorithm

- dan815

dude its all about designing an efficient searching algorithm

- dan815

why cant you see this bruh!! thats where the difficulty is

- Kainui

ok go on, sounds like rule based expert system. I've been watching through some algorithm design stuff for making artificial intelligence programs from MIT OCW

- Kainui

Maybe we should look at that and try to improve those

- dan815

u wanna start with someones code and project already?

- dan815

I thought we should try our own hand at making a searching program

- Kainui

sure, I don't care just as long as you pick something that'll be interesting enough until the end of the week

- dan815

okay :)

- dan815

lets do thsi then!!

- Kainui

I'm gonna heat up some coffee and brb, choose something and let's move into some other thing that's less laggy

- dan815

omg i am thinking of the craziest stuff yoo!!

- dan815

YO MAN OMG!

- dan815

i know how to program dictionaries BRO

- Kainui

HAhaha wat

- dan815

:)

- dan815

u really wanna see?

- Kainui

alright message me on skype sure

- dan815

okay :)

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