Mathematics - 2017-06-21 (page 5 of 11)

Typhon

04:00

"5 seconds later: '... and for all you know the universe was created 5 second ago' "

Semiclassical

If not, then I really don't care.

Typhon

@Dodsy true, but if done in fiction it would be hilarious

Dodsy

Denoting the universe a time "5" in this sense is arbitrary.

because I am 23.

meaning that this '5 seconds"

Semiclassical

And tomorrow I'm 30

Dodsy

is much slower than our "5 seconds"

Semiclassical

04:00

oh god oh god oh god

Typhon

fair point

arctic tern

also see

https://en.wikipedia.org/wiki/Philosophical_skepticism#Skeptical_hypotheses (seems to be missing the Boltzman brain hypothesis, but that's analogous to the brain in a vat one)

Dodsy

so, it might as well be 32 billion years.

Typhon

@Semiclassical BRAIIIN HURTS

@Dodsy I meant that a diety say within a void, clapped his hands and all the particles just materialized in the form of the current moment.

whatever

Dodsy

oh

Typhon

04:02

I was thinking of a funny line for the start of a story that would be incredibly ironic and you ignored me. XD

Dodsy

so then all of my past memories are false.

Typhon

@Dodsy yeah basically

PVAL-inactive

Anyway back to paths on a grid.

Typhon

NO

no more probability for tonight

please

3

Q: Are there such a thing as non-rectangular or non-Euclidean matrices?

I was thinking about surface geometry and spherical geometry in particular and I was wondering about whether or not there was such a thing as a matrix (as in vectors and matrices) in the shape of a sphere of numbers. Either the surface of a sphere or filled would count. What I mean is similar to ...

Semiclassical

heh, I like this quotation from the Omphal-whatever hypothesis from a rabbi

Typhon

04:03

^^^geometry anyone?

Semiclassical

"God essentially created two conflicting accounts of Creation: one in nature, and one in the Torah. How can it be determined which is the real story, and which is the fake designed to mislead us? One could equally propose that it is nature which presents the real story, and that the Torah was devised by God to test us with a fake history!
One has to be able to rely on God's truthfulness if religion is to function. Or, to put it another way—if God went to enormous lengths to convince us that the world is billions of years old, who are we to disagree?"

PVAL-inactive

I recently finished "The Witness", and would recommend it to anyone.

Dodsy

@PVAL-inactive do you have a solution??

PVAL-inactive

Most of that game is finding paths on a grid.

Typhon

@Semiclassical and my response is "well, if the guy came and took the time to tell everyone X, let's assume the literal Word of God is correct and not argue with the guy. Perhaps we're just looking at nature wrong."

Dodsy

04:04

@PVAL-inactive I've said this 4 times already, but we know that the grid needs to begin and end on a horizontal edge. For the case $n=1$ there is a 75% chance that there will be an edge at the beginning and a 75% chance there will be an edge at the end. We know that a straight line is ~46% chance for $n=1$, but we also see that it hinges on the vertical edge, which has a 50% chance of occuring.

What do you deduce from this information?

Typhon

anyway

i have an itch to do matrix fun stuff

Semiclassical

I think the point is that, given the universe itself would be created by God, all the various evidence we see for its age would itself be 'testimony' of God.

Typhon

anyone want to rip apart matrices and make other thingies?

PVAL-inactive

@Dodsy I don't think horizontal paths are super helpful here.

Semiclassical

@Typhon I forget what exactly you wanted.

Dodsy

04:06

no, I am not doing horizontal paths

I am considering the vertical edge in the middle

we also need to consider that the path needs to start and end on a horizontal edge.

Semiclassical

Were you looking for some kind of spherical arrangment of numbers, in the same way that a tensor would be a 'cube of numbers?

Typhon

@Semiclassical well it was human recorded and interpreted. So, there is always the potential for flaws. Written words don't have that possible error aside from reading interpretation. but i completely agree with your point. The universe should ultimately appear correct given a total and complete view of it at this current time.

Dodsy

so for $n=1$

there is a 75% chance that there will be a horizontal edge at the beginning and a 75% chance there will be a horizontal edge at the end, a ~46% chance there will be two horizontal edges connected, and a 50% chance that the vertical edge will be available.

Typhon

@Semiclassical yes. Or a surface of a sphere. Or a surface of really any surface. I was essentially asking if there are matrices out there that are neither rectangles or cubes but arbitrary surfaces and volumes.

Semiclassical

I suspect the problem you'll run into is that what's essential to a matrix (or a tensor) is its (multi)linearity.

Typhon

04:08

i see

Dodsy

Now, note that as $n\to\infty$ the chances of horizontal edges at the beginnings and ends approach 100%.

Typhon

yeah, but linear refers to line

Semiclassical

I don't see a way that a spherical arrangement could avoid that.

Not really.

Dodsy

but as $n\to\infty$ the chance of a straight horizontal path approaches zero.

Semiclassical

$M(au+bv)=a(Mu)+b(Mv)$.

That's linearity.

Typhon

04:09

why couldn't we have a linearity referring to... the lines of a sphere or other geometric surface if that sort of makes sense.

@Semiclassical I know, but it stems from the equation of a line (hence, linear)

Semiclassical

Eh. I remain dubious.

(The closest thing I know would be a "spherical tensor operator", and that should be understood as "spherical 'tensor operator'" i.e. a tensor operator which transforms in a certain way under rotations.)

Typhon

hmm fair enough

Dodsy

eh, idk why I am even thinking about this problem.

Typhon

@Dodsy because you read the words describing it and therefore they echo in your mind.

@Dodsy don't think about elephants

Dodsy

What's an elephant

Typhon

04:15

@Dodsy ffs

Semiclassical

something that never forgets (that the universe was created 5 seconds ago /s)

PVAL-inactive

@arctictern Yeah I'm pretty sure thats what you do.

Typhon

hahahaha

Dodsy

I wish I could just google the answer to this problem

so I could sleep at night.

mathworld.wolfram.com/BondPercolation.html

Typhon

@Dodsy then ask on MSE

that's what I do

Semiclassical

04:18

This page is fun re lattice percolation: web.mit.edu/8.334/www/grades/projects/projects14/PeterFedak/…

Typhon

i might ask a question to help me get the function I need to implement the hookh- oh sorry treegrabber.

:p

been wanting to get it working for some time but fail to make a simple multivariate step function.

Dodsy

basically reading a research paper to try to answer this question.

Typhon

I'd take the time to use words to describe it but I intend to use a diagram

@Dodsy well... never doubt the power of an answer on mse.

user76284

Abbreviating "if and only if" as "iff" in formal papers, yea or nay?

Dodsy

lmfao

PVAL-inactive

04:24

nay

Dodsy

yea

user76284

Vote now on your phones

Typhon

@user76284 Don't use either. Use "is equivalent to". Edit: BWAHAHA! I got a star!

2

PVAL-inactive

I've never seen a paper use iff

iff is shorthand for boardwork and notes.

Semiclassical

hmm. What I'm currently pondering is whether going to the dual graph may be useful

arctic tern

04:26

search "iff" on arxiv get "Your query resulted in too many hits, only 1000 hits are being displayed."

Jeff

@Typhon Cuz I can't stand the phrasing "if and only if". So using neither was my vote.

Dodsy

@Semiclassical @PVAL-inactive this was an interesting read nist.gov/sites/default/files/documents/itl/cxs/…

Semiclassical

@PVAL-inactive it goes down if you do " iff " on the full-text search: search.arxiv.org:8081/?query=%22+iff+%22&in=

Typhon

@Semiclassical actually, I believe the term would be hedonism. You live only in the moment.

Jeff

I cannot figure out how "only if" are words to mean "the left" implies "the right". It's baffling phrasing to me.

Typhon

04:27

@Jeff me too. Thanks for the vote.

well...

user76284

A only if B, that is, not B means not A, that is A entails B

Semiclassical

P only if Q means that, were Q false, then P would be false. Equivalently, if P is true it's because Q is true as well.

Dodsy

@Typhon that's not really hedonism.

Typhon

@Dodsy i was making a mild statement of a response.

:p

Semiclassical

If only because hedonism is all about experience. Can't have that if the universe only just existed

Dodsy

04:28

oic

Typhon

partially a joke

@Semiclassical true but then more of a reason to do something right now

Dodsy

where did astyx find this problem

and why didn't he share the solution.

Jeff

@user76284 OK, so I finally have my answer.... Can I still hate the phrasing.

PVAL-inactive

@Semiclassical If there are really only 290 papers doing it on all of arxiv that confirms my experience pretty accurately.

Typhon

@Semiclassical besides, believing that the universe came into being 5 second ago means that two hours from now the universe is 2 hours old.

Dodsy

04:29

after this, I'll know all about lattice theory and percolation theory

Jeff

@Semiclassical Same thing I said to @user76284. Can I still hate the phrasing?

Dodsy

and still not be able to answer this quesiton.

Semiclassical

@dodsy To fill in what I meant about the dual graph: Put a new vertex in each square, and draw an edge between vertices if there's an edge that's common to both cells.

Typhon

i think the answer is going to be 100% trivial.

ironic if the final probability of a connection is just... 1/2

Dodsy

@Semiclassical Continue.

PVAL-inactive

04:30

There's actually a few papers in which the specify they are using iff for if and only if.

Semiclassical

...huh. I think I have a thought.

Dodsy

@Semiclassical Divulge.

user76284

@Jeff Absolutely.

Sometimes I wish we had en.wikipedia.org/wiki/Speedtalk.

PVAL-inactive

and a few about the IFF

Jeff

@user76284 Whew

Typhon

04:31

@PVAL-inactive that doesn't make it right to do so

Dodsy

@Typhon it is unexpected.

Semiclassical

...no, damn, it doesn't work. hnnnnggh

user76284

There have been many attempts to create logic-based synthetic languages but none have taken off.

Typhon

@Dodsy fair enough

Jeff

@user76284 So is reading that book as difficult as reading a research paper?

Dodsy

04:32

No, I am agreeing with you

the fact that the answer is 1/2 is unexpected.

PVAL-inactive

Well if it's rare enough that you have to specify what it means, you shouldn't be using an abbreviation casually.

Semiclassical

What I was hoping to make use of was this:

Typhon

@user76284 so not programming?

damn

Dodsy

National Treasure Lemon Juice Scene

►

user76284

It's a logic-based language. Kind of like Lojban.

Typhon

04:32

@Dodsy best. scene. ever.

Semiclassical

The dual graph of the $n\times(n+1)$ square graph is an $(n-1)\times n$ square graph.

Typhon

@user76284 idk what that is. It's a verbal language though, not a programming one?

Dodsy

hmmm

Semiclassical

The dual to that is $(n-2)\times(n-1)$, and so forth.

Dodsy

what would a 0 x 1 graph look like?

user76284

04:33

Yeah it's meant to be a human language.

Semiclassical

Well, you'd stop by that point.

user76284

Lojban

Lojban (pronounced [ˈloʒban]) is a constructed, syntactically unambiguous human language, succeeding the Loglan project. The Logical Language Group (LLG) began developing Lojban in 1987. The LLG sought to realize Loglan's purposes, and further improve the language by making it more usable and freely available (as indicated by its official full English title, "Lojban: A Realization of Loglan"). After a long initial period of debating and testing, the baseline was completed in 1997, and published as The Complete Lojban Language. In an interview in 2010 with the New York Times, Arika Okrent, the author...

Dodsy

right.

Typhon

@Dodsy nothing... literally

Semiclassical

$1\times 2$ would be the lowest level.

Jeff

04:33

Here's a request for analogy help: A circuit (graph theory) is a closed trail. That is, it has all the properties of a closed walk (it starts and ends at the same vertex) and all the properties of a trail (no repeat edges). This works the same way that a square is both a rhombus and a rectangle. It has all the properties of both. Anyone got a better analogy I could use? A nice simple one, too. TY

Dodsy

which is basically just a 1 x 2

but flipped sideways

Semiclassical

I was hoping to bootstrap that, e.g. if there's a connection for the $n\times(n+1)$ graph then there's also one for the $(n-1)\times n$ graph.

Dodsy

meaning that instantly there would be a 75% chance that a connection would be made at $n=2$

maybe more.

oh yeah

an 87.5% chance.

oh I see.

Semiclassical

And so forth until you get down to the $1\times 2$ graph...heck, actually, down to the $0\times 1$ graph, understood as just two vertices with an edge between them.

In that case, the 1/2 fact is trivial.

Dodsy

that'd be nice.

I wish that was the problem

PVAL-inactive

04:36

@semiclassical I don't think the numbers of paper given by arxiv full text search are accurate.

Semiclassical

@PVAL-inactive I certainly agree on that.

Dodsy

"there is an edge, with 50% chance that there will be an egdge, what is the chance that there is an edge"

PVAL-inactive

if and only if only returned like 360

Semiclassical

I was just using it as a measuring stick.

Typhon

hypothetical thought, what if assigned numerical codes to types of functions? And then we wrote them using postfix notation: a + b becomes: <1,0,0>. Is there any way this could make integration easier?

Dodsy

04:37

hmm.

Semiclassical

That seems kinda like asking whether working in machine code would make game design easier, tbh.

PVAL-inactive

to clarify I don't think its even an accurate search of arxiv's database.

Typhon

@Semiclassical yet we use adders to do addition....

Semiclassical

yeah, and addition is really damn simple.

Typhon

we don't write it from boolean operators in code

Semiclassical

04:38

but it's also not clear what you mean by 'integration' here. Symbolic? Numeric?

Typhon

@Semiclassical true but integration is generally difficult. My thought was to somehow write differentiation as a vector operation.

integration means finding the function that is the integral

Semiclassical

So symbolic.

Typhon

numeric is just approximation

Dodsy

What if we made a machine

Semiclassical

"just"

Dodsy

04:39

that interpreted 1's and 0's

Semiclassical

There's an algorithm for symbolic integration, though.

Dodsy

as information

Semiclassical

Risch algorithm

In symbolic computation (or computer algebra), at the intersection of mathematics and computer science, the Risch algorithm is an algorithm for indefinite integration. It is used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch...

Dodsy

and then could perform difficult tasks using that information quickly.

Typhon

@Semiclassical what if one could write differentiation as a vector operation. Then wouldn't finding the inverse of that function reveal an integral for all derivatives of elementary functions?

Semiclassical

04:40

@Dodsy Man, those machines would be have to be huge.

PVAL-inactive

There is also an algorithm for telling if a function has an integral expressable in elementary functions

Semiclassical

@PVAL-inactive Isn't that also the Risch algorithm?

PVAL-inactive

and that algorithm is implemented in code so it must have a reasonable runtime.

Dodsy

@Semiclassical I'd probably put a 13'' screen on it.

Typhon

@Semiclassical I'm not talking about machine programming. I'm talking about a direct numeric operation. Literally a vector function.

PVAL-inactive

04:40

it might be.

I didn't know the name.

Typhon

@PVAL-inactive ROFLMAO

Dodsy

Algorithms are cool.

Typhon

@Dodsy true, but a literal function of sufficient simplicity is preferable.

Semiclassical

@Typhon I have no idea what this could mean, so I guess I'll say: If you can do it, great.

Typhon

@Semiclassical I meant like translating functions like i said into vectors and then finding an operator that maps these vectors to the corresponding derivative vectors

possibly a basic vector function

or maybe a 100 term behemoth

XD

PVAL-inactive

04:43

I suspect that such a matrix would be a complicated matrix with a giant dimension, and multiplying it by a vector would take a long time.

Semiclassical

Plus, functions have infinitely many degrees of freedom.

Dodsy

Functions are pretty lit.

Typhon

meh

free thought

Semiclassical

You could write them as linear combinations of basis functions and think of those as vectors....but people already do that. That's what Fourier analysis is.

Dodsy

Semi

did you ever do putnam exams

Semiclassical

04:44

nah.

Typhon

i meant like how there is postfiix notation

Dodsy

But you're brilliant mate.

Semiclassical

there were a few regional things, but not putnam

Dodsy

Strange.

Typhon

just give all the elementary functions a numeric code

and then write functions as vectors. XD

Dodsy

04:44

A guy from waterloo did really well on putnams

Typhon

(i suppose general weird curves wouldn't apply)

and the inverse wouldn't exist for all vectors

or... rather all the vectors that are valid functions.

Semiclassical

I'll stick with my "if you can do it, great" line.

Typhon

@Semiclassical once again, free thought

Semiclassical

As it stands I see no reason why this is a useful description of the problem.

Typhon

making small talk

Dodsy

04:46

Daniel Spivak

Semiclassical

@Dodsy Actually, quick question: Are horizontal edges allowed at the top/bottom?

Dodsy

yeah

Semiclassical

dang.

Typhon

having trouble with my game indiedb.com/games/block-builder

anyone here good with step functions?

Dodsy

Daniel Spivak is a good name for a mathematician

placed top 5 and is a putnam fellow

I'd go to Uwaterloo if it didn't look like a boring place to live.

Semiclassical

04:50

@Dodsy Here's an instance of the dual graph idea

Dodsy

please share

interesting

PVAL-inactive

@semi neither has a path though.

Dodsy

yeah

PVAL-inactive

or never mind

Dodsy

you see one?

PVAL-inactive

04:52

I guess the dual is shifted one to the right

Dodsy

left or right

Semiclassical

Well, I'd ideally want to take the dual graph to really only be the lattice parts which don't connect to the boundary

so a 4-by-5 graph rather than the original 5-by-6

Dodsy

The interesting thing is that I'm pretty sure astyx solved this in like 15 minutes

Typhon

Suppose the top left of the grey square is 0,0

the function increases as the squares go out

each one goes higher by 1

so grey = 0

red = 1

Semiclassical

Typhon

04:55

dark red = 2

etc.

does anyone know a function for this?

Semiclassical

This version is the problem for me as far as the dual graph goes.

Typhon

I think it is some combination of x +y and floor and absolute value but it eludes me for some reason

Dodsy

Here's what astyx said before he solved it

"Yeah but all possible outcomes are obviously equiprobable, so partitionning the set of outcomes (winning and losing ones) and making a bijection between those would solve the question"

Semiclassical

@typhon should be $|x-3|+|y-3|$.

Typhon

um...

Semiclassical

04:58

The center is (3,3) is it not?

Typhon

that's not a step function...

no

Semiclassical

...woops

Typhon

"Suppose the top left of the grey square is 0,0"

Semiclassical

hrm.

Typhon

and the squares are 64*64

and it is a step function

Semiclassical

04:58

I misread that, yeah.

Typhon

basically it determines the distance relative to the player in spaces

Semiclassical

Not sure, then.

I think some combinations of absolute values with floor/ceiling functions would work.

Typhon

which I need to implement a cough hookshot cough treegrabber.

@Semiclassical that's what I thought as well

Semiclassical

tree-nabber

Typhon

lol

@Semiclassical was thinking of |floor(x+y)|

Dodsy

04:59

yeah he asked the question at 8:16

Transcript for

$Mathematics$ Mathematics