Mathematics - 2017-01-22 (page 2 of 5)

TheGreatDuck

02:00

@AkivaWeinberger sin and cos are good for finding the height and base of a triangle. Like, if you want to build a ramp.

or a pyramid

Socrates

the pyramides astonish me in general

such huge boulders, it is really unbelievable

Akiva Weinberger

If there's a will, there's a way.

Socrates

I like the idea of using wet wooden triangles to blast boulders out of a cliff(?)

(during night)

TheGreatDuck

just wooden spears

in general

Akiva Weinberger

For modern stuff, they're important for computer graphics, like in 3D video games

TheGreatDuck

02:04

triangle is probably just referring to simple things

Akiva Weinberger

Sines, not pyramids

TheGreatDuck

@AkivaWeinberger i know first hand they are not used in 3D model rendering

Akiva Weinberger

Really?

TheGreatDuck

we just use similar triangles to project to a plane

and then clip and rasterize

Akiva Weinberger

Oh

TheGreatDuck

02:05

(in the naive sense anyway)

oh wait no!

you need sine and cos to calculate rotation matrices to rotate models and the camera.

Socrates

aren't polygons exactly that? a bunch of triangles?

Akiva Weinberger

There we go ^^

(@TheGreatDuck)

TheGreatDuck

@Socrates but chunks of wood are not polygons. :-)

Akiva Weinberger

Chunks of wood are a bunch of tetrahedra :P

TheGreatDuck

really? -_-

what is an atom?

Socrates

02:07

a model!

Akiva Weinberger

Well, it's approximately a bunch of tetrahedra, anyway

TheGreatDuck

@Socrates 3D models are made out of triangles in space. you are correct.

Simply Beautiful Art

To me, an atom is...

well, it was supposed to be the smallest thing

that could not be broken into something smaller

Socrates

@TheGreatDuck well, I'm certainly not the best one to discuss 3d graphics, as I'm in love with isometric :)

today an atom would equal to maybe a string?

(if we continue the original idea)

Simply Beautiful Art

Yeah, probably

TheGreatDuck

02:10

let's not get into string theory

Akiva Weinberger

> In a paper written by Marlow Anderson and Todd Feil, linear algebra is used to prove that not all configurations [of Lights Out] are solvable and also to prove that there are exactly four winning scenarios, not including redundant moves, for any solvable 5×5 problem.

O_O

My proof is wrong, then

Simply Beautiful Art

lol

Paul Plummer

Say we have hyperbolic surface, and simple closed curves $a,b$ and $\bar a, \bar b$ and the pairs are in minimal position, with $a,\bar a$ homotopic, and $b, \bar b$ homotopic. I really think there should be homotopies $F,G$ which if ran at the same time take $a$ to $\bar a$, $b$ to $\bar b$, and $a_t=F(,t),b_t=G(,t)$ are in minimal position the whole way. I am having a hard time justifying this though. It feels obvious in the universal cover, but I haven’t proved it.

Akiva Weinberger

Sorry, I don't know this subject

2

Simply Beautiful Art

@AkivaWeinberger What are your hours of business?

Akiva Weinberger

02:15

??? @Simply

What is "minimal position" @PaulPlummer

I know what homotopies are

TheGreatDuck

apparently we are now tutors.

XD

jk

Akiva Weinberger

I'm also not entirely sure what a hyperbolic surface is. Constant negative curvature?

Simply Beautiful Art

I honestly feel that way too @TheGreatDuck

Akiva Weinberger

Or negative total curvature?

TheGreatDuck

the former

hyperbolic geometry

those surfaces

Simply Beautiful Art

02:17

@AkivaWeinberger When are you online?

TheGreatDuck

hyperboloid of two sheets either half

hyperboloid of one sheet

Akiva Weinberger

@TheGreatDuck That's not constant curvature

TheGreatDuck

well

Akiva Weinberger

The pseudosphere is constant negative curvature

TheGreatDuck

that's ones that are fulfilling of hyperbolic geometry.

Akiva Weinberger

02:17

though it has a boundary

TheGreatDuck

pseudosphere?

you mean the poincare disk?

Akiva Weinberger

No

Paul Plummer

Constant negative curvature. Minimal position for curves is when we have that out of homotopic representatives for those curves they have the fewest number of intersections. This is equivalent to the curves being transverse and having no bigons @AkivaWeinberger

Akiva Weinberger

Bigons?

Oh, like they cross each other and "uncross" each other?

TheGreatDuck

bigons be bigons

Paul Plummer

02:19

Yah

Akiva Weinberger

In such a way that the area between them is homeomorphic to a disk?

Also I'm guessing we don't need this thing to be embeddable in $\Bbb R^3$. So, like, the hyperbolic plane and its quotients count

and the double torus with the right metric

(which actually is a quotient of the hyperbolic plane I think)

In any case, I understand the problem now, but I have no idea how to prove it

especially since I don't know the subject (except for a few definitions)

I think @TedShifrin knows, though.

There's a bunch of users who I think might be more helpful, but none of them are online right now @PaulPlummer

Paul Plummer

Yah, I don't really care about the metric, I just care about the topology at the moment, and maybe the geometry would make this easier to prove, so I figured I would mention it.

Akiva Weinberger

I know some algebraic topology (from reading the first two chapters of Hatcher), so I know what homotopies and universal covers are, but Hatcher doesn't cover anything about minimal position.

Kasmir Khaan

@TedShifrin Hi Ted

Akiva Weinberger

(At least, I think he doesn't.)

Paul Plummer

02:27

I don't think he does. Basically I want to show some construction is independent of the homotopic representatives I choose, and this minimal position business is part of the construction.

Akiva Weinberger

What course is this from, if I may ask?

Or what subject?

TheGreatDuck

i gotta go pee

Akiva Weinberger

…So do that

Paul Plummer

I think I have a way around my above question, but I would prefer to know that the above is true, and how to prove it.

TheGreatDuck

brb

Paul Plummer

02:30

I am reading a paper, Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs. I understand most of the paper, but I can't quite get this crucial detail rigorously. It is part of geometric group theory, low dimensional topology @AkivaWeinberger

The detail is actually about arcs, but I figured a proof for simple closed curves would extend to arcs

Kane

Hey does anybody know why the infinite series from 1 to inf: sin^2(1/x) converges?

I've managed to work out that sin^2(x)/x^2 <= sin^2(1/x) but I'm stuck here

TheGreatDuck

sure it does?

and it's not just a matter of how you looked at it?

Akiva Weinberger

@TheGreatDuck Periodicity can't come into it. $0<1/x<2\pi$, so it all takes place in one period

TheGreatDuck

i knew that. I totally knew that. This isn't sarcasm. You are not passing go. You are not collecting 200 rep. Move along. Move along.

XD

Akiva Weinberger

@Kane Hint: $\sin(x)<x$ for positive $x$

Simply Beautiful Art

02:39

DANG IT @AkivaWeinberger Stealing my stuff

TheGreatDuck

hey

Akiva Weinberger

@TheGreatDuck Cálmate

TheGreatDuck

as long as you guys aren't solving diff eq I'm good.

XD

Akiva Weinberger

So, here's why there is no general solution for Light's Out on a $5\times5$ board (per Wikipedia).

The parity of the nondiagonal buttons doesn't change.

By which I mean, whether or not there's an odd or even total of buttons that are on here doesn't change by pressing buttons:

TheGreatDuck

light's out?

Akiva Weinberger

02:45

\begin{matrix}O&X&X&X&O\\X&O&X&O&X\\X&X&O&X&X\\X&O&X&O&X\\O&X&X&X&O\end{matrix}

TheGreatDuck

oh those turn off all the lights but the adjacent ones toggle as well puzzles?

Kane

@AkivaWeinberger hmm, yeah I know that fact but I'm having trouble applying it to the part of the problem I'm stuck on.

So I can use the fact that sinx < x somehow 'at this part of my problem' sin^2(x)/x^2 <= sin^2(1/x)?

Akiva Weinberger

First, if you don't have LaTeX enabled, there's a link on the top-right that does that

But plug in $1/x$ to the inequality to get $\sin(1/x)\le 1/x$

and then square both sides to get $\displaystyle\sin\left(\frac1{x^2}\right)\le\frac1{x^2}$

Kane

@AkivaWeinberger I'll try get latex going first

Akiva Weinberger

It renders all of the code we've been typing, turning it into math formulae

so $a^b$ turns into an actual exponent (with the b smaller and raised)

@TheGreatDuck Yeah. I was discussing it a while ago

There's another (linearly independent) set of buttons whose parity is invariant of button presses

Everything but the second and fourth columns, and the middle row

TheGreatDuck

02:51

@AkivaWeinberger random question. What would be the best way to define an alternate version of the derivatives. Examples, axiomatic identities, or what?

Akiva Weinberger

You can see them in the "gameplay" section here

TheGreatDuck

i know what they are

i've done quite a few

Akiva Weinberger

The infinitesimal versions are nice

Like, in the ring $\Bbb R[\epsilon]/\langle\epsilon^2\rangle$, we can define it so $f(x+\epsilon)=f(x)+f'(x)\epsilon$

(assuming there's a nice way to define $f$ on that ring)

What that ring is, is essentially it's the set of numbers of the form $a+b\epsilon$ where we define $\epsilon^2=0$

and leave $1/\epsilon$ undefined

There's a name for it, let's see if I can find it

Kane

@AkivaWeinberger ah.... related question, I thought this was true:

$\sin x \leq x$
\frac{1}{\sin x} \leq \frac{1}{x}
\frac{1}{\sin^2 x} \leq \frac{1}{x^2}

Is that incorrect?

Akiva Weinberger

Ah, dual numbers @TheGreatDuck

Kane

02:56

hmm, latex fail

Akiva Weinberger

You need dollar signs, \$like~this\$

$like~this$

@TheGreatDuck Try calculating $p(x+\epsilon)$ for some polynomials

TheGreatDuck

why?

Akiva Weinberger

@Kane It is incorrect; if $a$ and $b$ are both positive and $a\le b$, then $\frac1a\color{Red}{\ge}\frac1b$.

The inequality flips.

More generally, if they're the same sign the inequality flips, and if they're opposite signs (one positive, one negative), the inequality stays the same.

@TheGreatDuck To see how the definition of derivative I just gave works

It's also cool seeing how the product rule comes so easily from it

(The chain rule comes less easily)

TheGreatDuck

well I'll look later

does it say anything about step functions?

like whether they have a derivative undefined at points?

Kane

@AkivaWeinberger Hold on I better write this again I think I made a typo.

$\sin x \leq x$
$\frac{1}{x} \leq \frac{1}{\sin x}$
$\frac{1}{x^2} \leq \frac{1}{\sin^2 x}$

Akiva Weinberger

03:01

I guess you'd need to specify that $\epsilon>0$ for the step functions to be defined, but yeah, the definition would work, and it would be undefined at the discontinuous points @TheGreatDuck

Kane

@AkivaWeinberger does the inequality flip again when I square both sides?

TheGreatDuck

hmm

Akiva Weinberger

@Kane This is correct

@Kane It does not flip

TheGreatDuck

I'm trying to make a derivative where there is a rule saying that the derivative of step functions is universally 0.

Akiva Weinberger

I mean, it doesn't if they're both positive @Kane

If they're both negative, it does. If they're opposite signs, then we don't have enough information.

This has to do with where the function $x^2$ is increasing.

(If they're opposite signs, it might even turn into an equality: $-2\le2$, but $(-2)^2=(2)^2$.)

Kane

03:05

@AkivaWeinberger yep got it. But for this question, the proper way to do it is like this:

$\sin k \leq k$

let $k = \frac{1}{x^2}$

$\sin \frac{1}{x^2} \leq \frac{1}{x^2}$

Akiva Weinberger

@Kane Yeah

Well, wait

I thought the question was about $\sin^2(1/x)$

not $\sin(1/x^2)$

Kane

@AkivaWeinberger isn't it the same thing?

Akiva Weinberger

No.

Kane

@AkivaWeinberger Lol, I'm so bad at this

Akiva Weinberger

You plug in $k=\frac1x$, and then square both sides

Kane

03:09

@AkivaWeinberger so $\sin^2 \frac{1}{x} = (\sin \frac{1}{x})^2$

Ahhhh, OK I see

Akiva Weinberger

Yeah. It's weird and kind of annoying notation, but yeah

(You're not annoying, the people who made that notation standard are annoying.)

Kane

@AkivaWeinberger Lol, yep I understand what you're saying :). You have been a great help, thankyou so much!

Akiva Weinberger

You're welcome

Sophie

03:46

someone is asking about finding $\int \frac{8x^2+6x+4}{x+1}dx$ without using $\int (f+g)=\int f + \int g$

math.stackexchange.com/questions/2108231/…

Semiclassical

@Sophie It's been removed.

Akiva Weinberger

(((xy)z)(x((xz)x))) = z

Apparently, that one axiom describes Boolean logic

"xy" means at least one of x and y is false; it's known as the "NAND" operator.

All other logical operators can be defined in terms of it. For example, xx means $\lnot x$. (xy)(xy) means $x\land y$.

And the above axiom is the shortest possible axiomatization of logic.

It's called the Wolfram Axiom.

Ramanujan

04:25

Yeppiie,iam now an established user on mathematics $\Huge{\text{:-)}}$

TheGreatDuck

@AkivaWeinberger look up "weak derivative"

now we need solutions to equations built from weak derivatives

I think this in and of itself generalizes my whole idea in a nutshell :D

Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e. to lie in the Lp space L 1 ( [ a , b ] ) {\displaystyle \mathrm {L} ^{1}([a,b])} . See distributions for an even more general definition. == Definition == Let u {\displaystyle u} be a function...

Akiva Weinberger

04:47

Wow

@TheGreatDuck Notice that, in their weak derivative of $|x|$, the value at $x=0$ is unimportant; any function that differs from it at just one point is also a weak derivative

More generally, any function that differs from it at a null set is also a weak derivative

TheGreatDuck

@AkivaWeinberger interesting. I assumed they used the average.

ooh

perhaps not what I want then

i made a question

someone will answer whether it's right or not

if it is, then I can fill in all the holes and ask a real question.

then again, I think I might wait so I can make it as a Q&A I can prove.

@AkivaWeinberger it's amazing how it took me a year and 2 months to find that thing.

if it is the right thing. XD

Socrates

05:04

@AkivaWeinberger and I thought I am the only one.

Kaj Hansen

Bonsoir chat

Socrates

@KajHansen Salut mon frer

TheGreatDuck

@KajHansen are you familiar at all with "weak derivation"?

Kaj Hansen

Never heard of it @TheGreatDuck

TheGreatDuck

hmm

know that weird differential equations stuff I was talking about?

Socrates

05:09

@TheGreatDuck how about: you try to explain it to us?

TheGreatDuck

@Socrates i was actually trying to see if maybe he could answer a question I had. XD

Kaj Hansen

Est-ce que tu sais francais @Socrates ?

Akiva Weinberger

Type "uncle's uncle's uncle" into Wolfram Alpha

Socrates

@KajHansen a peut

Kaj Hansen

@Akiva, will do. You should type "cubic light year of jello" into wolfram

Do you encounter many French speakers IRL @Socrates ?

Socrates

05:13

@KajHansen not many that I know of.

Kaj Hansen

haha, "great granduncle"

Akiva Weinberger

See, this is the sort of thing that makes me think I'm not using Wolfram Alpha to its fullest potential.

Socrates

wolfram alpha is just a tool

Akiva Weinberger

And I don't think I'm using it to its fullest potential.

Kaj Hansen

hahaha

Akiva Weinberger

05:14

Brb, eating a cubic light year of jello

Socrates

try to use a hammer to its full potential

Kaj Hansen

You can type other stuff in that @AkivaWeinberger. Pudding etc

Akiva Weinberger

The difference here is that a hammer is not eight million lines of code long.

There's a difference in scale.

Socrates

@AkivaWeinberger can you actually code a hammer, such that it appears in your hands?

Kaj Hansen

@AkivaWeinberger, "$2.57 \times 10^{51} \% $ daily value of zinc :D

Sophie

05:17

@AkivaWeinberger is there a name for my mother's husband's mother?

Akiva Weinberger

Just a sec

Kaj Hansen

LOL

Socrates

here is how I see it: a hammer helps me to punch a nail in the appropriate place. Wolfram Alpha can never be as useful as a hammer in that regard.

Akiva Weinberger

Oh, derp.

Lol, sorry

Kaj Hansen

Grandmother

Sophie

05:18

wolfram alpha just says grandmother

Akiva Weinberger

Yup

Sophie

just no

Akiva Weinberger

I mean, it assumes the parents are married

Sophie

we can call her $\zeta$

Akiva Weinberger

@Socrates No, the hammer punches the nail; you punch it in the appropriate place.

Also, I think the verb is actually just "to nail"

TheGreatDuck

05:20

just don't put screws in the walls

you'll "screw it all up"

Socrates

@TheGreatDuck :-D

TheGreatDuck

D-:

Akiva Weinberger

By the way, do you guys all know about Turing machines?

Socrates

I heard of the concept, but I wouldn't say I know about them.

Akiva Weinberger

Apparently, there's a 2-state 3-color universal Turing machine

TheGreatDuck

05:22

turing machines are just programs

the concept of a computer process is a turing machine

Akiva Weinberger

It's more specific.

TheGreatDuck

granted, turing machine might be a broader term

Akiva Weinberger

Essentially, you have an infinitely long tape ("memory")

TheGreatDuck

oh

no virtual memory space?

Akiva Weinberger

Like, an infinite row of squares, called cells, in which you can write.

TheGreatDuck

05:23

yeah

infinite memory indices

Akiva Weinberger

And your index finger, or a mechanical reader, is somewhere on this tape.

TheGreatDuck

I'm a computer science major. no need to talk down to me.

Akiva Weinberger

It/you can only read one cell at a time.

TheGreatDuck

the index finger reads a command, interprets and then executes it, right?

and then it repeats moving down to the next command, right?

Akiva Weinberger

No, not really

TheGreatDuck

05:24

oh

Akiva Weinberger

but sort of

TheGreatDuck

:p

that's the idea of the von neumann computer achitecture

Socrates

but it's ok, as i'm no major

TheGreatDuck

basically the processor has 32 or so local slots that can be manipulated

Akiva Weinberger

The head can be in one of several "states" (the amount depends on the Turing machine), and it can write one of several "symbols" on the cells (again, depends on the Turing machine)

TheGreatDuck

05:25

and the finger fetches a command to be interpretted

Socrates

also, great self esteem, if you can't take "talking down to you"

Akiva Weinberger

@TheGreatDuck This all predates that.

It's meant to be theoretical, not as a blueprint for building these.

TheGreatDuck

@Socrates it felt like akiva was being awkward and having trouble about it so I told him no need to try and simplify it. :-D

I can take it just fine.

Akiva Weinberger

Depending on the state of the head and the symbol in the square, it will either erase the symbol, replace it with a new one, or do nothing; it will change state; and it will move either one cell to the right or one cell to the left

@TheGreatDuck I apologize, I wasn't trying to talk down or anything

Socrates

that's awkward. simplyfiying for a knowing one, is like complexifiying for a beginner, sometimes.

Akiva Weinberger

05:27

And then it repeats with the new cell.

TheGreatDuck

@AkivaWeinberger no need to apologize. I thought you were having trouble and so I wished to spare you the trouble of simplifying it. :D

Akiva Weinberger

Here, they use colors rather than symbols, but it's the same idea:

wolframscience.com/prizes/tm23

It was proven to be universal

meaning it's as powerful as any computer, as powerful as any programming language

TheGreatDuck

intriguing

reminds me of automata

never learned them

but the diagram looks similar

Akiva Weinberger

You can see the six rules, depending on the state of the head (also known as reader) and the color.

TheGreatDuck

@AkivaWeinberger I see. It's like the barest bone version of machine code.

Socrates

05:29

show it the most intuiitive that the probability for throwing at least 2 heads out of 3 tosses is equal to one half.

Akiva Weinberger

The diagram on the left essentially shows the tape progressing through time as you go down. It's starting with a blank tape, but you could start it with things on the tape already.

That would be the "input".

TheGreatDuck

0, 1,1,1,2,2,2,3 are all the number combinations

half are >= 2

therefore

1/2 probability

clear?

Socrates

for me yes, but I'd have to lobotomize myself to give a good judgement.

Akiva Weinberger

The reason it's "universal" is that any Turing machine could be simulated by this one, just by varying the input.

TheGreatDuck

@AkivaWeinberger it's neat, but to be honest. I'm not really in the mood to read that right now. I'll probably look at it later if that's ok. :-D

Interesting

Akiva Weinberger

05:31

OK

TheGreatDuck

@Socrates well. Have you taken any abstract math classes?

here's a good way to look at proving things

Akiva Weinberger

Turing was the one who first showed that universal Turing machines exist. This means that you don't need to build one machine to multiply numbers for you, another machine to calculate digits of pi, another machine to play Tic-Tac-Toe, etc

You just need to build the universal Turing machine

and it can calculate anything you want it to.

TheGreatDuck

the strength of a proof depends on writing style, audience, and what you accept as previous knowledge

Akiva Weinberger

That's why computers are possible.

TheGreatDuck

@Socrates I could insist you prove all of probability to prove it has chance of 1/2

but that's absurd

Akiva Weinberger

05:33

So, the innovation with the specific universal Turing machine I showed you is that it's so small!

TheGreatDuck

showing all the cases of coin flips is reasonable proof

Akiva Weinberger

Just three colors and two states!

TheGreatDuck

@AkivaWeinberger so are those six things the operators?

Socrates

@TheGreatDuck I think that a proof is only nice if you let the audience do it basically, with you being a guide.

but that's only my opinion

TheGreatDuck

@Socrates i'm talking about written proofs

and believe it or not

do you feel an urge to write "we" rather than "I" when doing math?

Akiva Weinberger

05:34

The six things are the six rules. If the head is in state "top" and the cell is orange, rewrite the cell to yellow, move left, and go to state "top"

That's the first rule

TheGreatDuck

@AkivaWeinberger oh, so it's not like operators then.

Socrates

@TheGreatDuck mmh, I think about that

TheGreatDuck

like "add this memory index to this memory index"

Akiva Weinberger

Because it's so small and already universal, it means that universal "computers" probably arise all the time in nature.

TheGreatDuck

@Socrates "we" is the proper pronoun because the idea is that in a proof both you and the reader are essentially working through it together. Of course, you did all the true work. They just have to read it and understand. :D

Akiva Weinberger

05:36

Yeah, you write "we" in proofs

TheGreatDuck

@AkivaWeinberger I know the CS teacher once made a random remark about dna being a von neumann architecture complete with an identifiable heap space, function space and yadda yadda yadda

though some things probably vary

Socrates

I really don't know.

TheGreatDuck

@Socrates fair enough. Anyway, your view on proofs is the appropriate view.

you just never learned it formally. You've developed the right opinion through experience.

Socrates

I like "proofs" that give me the tools, and let me hammer.

TheGreatDuck

well...

Akiva Weinberger

05:38

Von Neumann architecture is engineered

TheGreatDuck

a proof is supposed to stand on it's own three feet so it cannot tip over

Akiva Weinberger

The idea is that maybe universal computers exist everywhere in nature by chance

TheGreatDuck

and it's just an explanation as to why something is true

@AkivaWeinberger fair enough, but dna is an example of not only a universal computer, but one coincidentally using the von neumann architecture. Chew on that.

we just don't know the meaning of everything in it obviously

but that's no different than people unable to decompile some random program

Socrates

a banana has 50%(?) the same DNA as a human

what does this say

. reddit.com/r/shittyaskscience/comments/2lghoq/… :-D

Kaj Hansen

It means cells have a lot of the same stuff regardless of the species @Socrates. All cells have mitochondria that work essentially the same way, plus DNA replication, ribosomes, etc. All are the same mechanism from organism to organism.

Secret

05:44

Monocercomonoides

Monocercomonoides is a genus of flagellate Excavata belonging to the order Oxymonadida. Monocercomonoides species have been discovered living in the guts of small mammals, snakes, and insects. The genome of Monocercomonoides has approximately 75 million base pairs (75 Mbp), with 16629 predicted protein-coding genes. Many excavates lack "classical" mitochondria. Oxymonads lack true mitochondria and Golgi apparatus. Monocercomonoides has been characterized as the first example of a eukaryotic organism devoid of mitochondria. Its genome contains no mitochondrial DNA (mtDNA), and no genes for cardiolipin...

counterexample to the above statement

Kaj Hansen

Ok, sure @Secret

But most organisms this is true

Secret

indeed

I think if a biologist became a mathematician mid way, he/she might be very good at counterexamples

Kaj Hansen

haha, counterexamples are great. They're pretty illuminating in math

Akiva Weinberger

Fun fact 1: Every nonnegative number is the sum of four squares

Fun fact 2: This might be the easiest way to define "positive" if your structure is $(\Bbb Z,+,\times)$

Kaj Hansen

Squares of complex numbers?

Akiva Weinberger

05:48

A number $x$ is defined to be nonnegative if $\exists a,b,c,d:a\times a+b\times b+c\times c+d\times d=x$

@KajHansen This is in $\Bbb Z$

Kaj Hansen

Ohh, I thought you wrote every negative number

Akiva Weinberger

Like, if you can't use $\le$, so you can't just define it to be the set of numbers for which $0\le x$

So you could only use $+$, $\times$, $=$, and logic symbols

TheGreatDuck

@KajHansen plus in terms of memory within dna (there are portions that change like as in a program) they all have the same core components and structure like how computer programs are all very similar in architecture.

Kaj Hansen

I could see that

Socrates

a cynical view on beings would be that all beings are only servants of their genes.

TheGreatDuck

06:03

...

i'm just saying the things controlling cell growth and whatnot

im not cynical

Socrates

That makes one wonder why some individuals do things that are probably killing themselfes. Like working in Tschernobil or smoking cigarettes.

Kane

Guys, any ideas on how I can show this converges:
$\sum_{x=1}^{\infty } \frac{\sin ((2x-1) \times \frac{\pi }{4}) }{2^{x}}$

I'm thinking comparison test for series

or maybe ratio test

Kaj Hansen

Yeah, the numerator is bounded by $\pm 1$

So you can show absolute convergence, compared with $1/2^x$

Kane

@KajHansen ahh, good idea. Thanks Kaj

Kaj Hansen

Glad I could help :D

TheGreatDuck

06:17

does sin(x) converge or diverge?

Kaj Hansen

the latter

TheGreatDuck

it doesn't increase without bound though.

Kaj Hansen

diverge just means "doesn't converge"

N1ng

The sin jumps between $$\pm\frac{1}{\sqrt{2}}$$?

TheGreatDuck

@KajHansen smart man. ;)

i was pointing out that being bound by -1 and 1 is not enough to prove convergence.

Kaj Hansen

06:24

That's true, but the fact that it hits $1$ and $-1$ between every $2\pi$ interval is

Socrates

what is the word for divergent, but bounded by below and above? opposed to divergent and always increasing.

or what is the word for the latter

TheGreatDuck

oscilating?

Socrates

ah, yeah i guess

@KajHansen I want to eat raw eggs for all meals, together with vegetables

Kaj Hansen

Why raw ?

Socrates

I strongly believe that if something isn't digestable raw, it's not supposed to be eaten.

I play around screwing animal products all together and only eating vegetables, but well.

Kaj Hansen

06:41

I mean, it is digestible raw

I just imagine the culinary experience being more enjoyable with cooked eggs lol

Socrates

Certain kinds of raw fish are nosebleed cuisine.

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$Mathematics$ Mathematics