Mathematics - 2017-07-18 (page 3 of 5)

Typhon

17:00

integer coefficients as in the ordinary integers

1,2,3,4...

@PVAL-inactive I know that word, but it is not helpful here

Eric Silva

i mean it is though

Typhon

...

I'm not working with the set of algebraic integers.

not all of them

Alessandro Codenotti

ok so $-\sqrt[3]{2}$ satisfies the polynomial $X^3+2$. Does it satisfy a second degree one?

Typhon

@AlessandroCodenotti exactly

Eric Silva

if you knew more stuff about algebraic integers you would know the answer to your question

Typhon

17:02

@EricSilva ok, but I'm only asking if it is true. I don't want a proof.

@AlessandroCodenotti yes.

but wait...

Alessandro Codenotti

depending on what you know about field extensions and minimal polynomials the question I asked you above can be either very easy or not easy at all so think about it

Typhon

sqrt[3]/2 isn't a solution to a second order polynomial...

PVAL-inactive

The main point is the fact that the minpoly always divides any polynomial a root satisfies.

Typhon

@AlessandroCodenotti no! That's not an example.

ugh

Eric Silva

@Alessandro adjoining that doesn't give you a quadratic extension

Typhon

17:03

A quadratic field is any field Q[x] such that x^2 + ax + b = 0, where $a$ and $b$ are integers. The elements of any quadratic field are quadratic rationals.

@AlessandroCodenotti the cube root of 2 isn't a quadratic integer or a quadratic rational....

Eric Silva

@Typhon the thing you just defined isn't a field

oh i guess it is

Typhon

@EricSilva yeah it is. Do you need a proof?

Eric Silva

yeah never mind

Typhon

:-)

Alessandro Codenotti

Oh, right, give me a moment to come up with a better one

Typhon

17:05

@EricSilva and I didn't just define it. That's a quote from 10 minutes ago.

@AlessandroCodenotti alright

brb

Mike Miller

I am sleepy

Akiva Weinberger

Let's make lots of noise

Balarka Sen

I watched shitvideos for so long yesterday that I ended up messing up my sleep schedule.

Daminark

@Mike Did you not sleep too well? Or is it like, general fatigue?

Alessandro Codenotti

Actually I think the answer to your question might be affirmative @Typhon

Fargle

17:10

I Am So Great

►

Eric Silva

@Typhon You "just" defined it again, don't be rude.

Akiva Weinberger

@Typhon Whatever the monic polynomial is, factor it

Daminark

@Balarka Washington

And gg

Akiva Weinberger

Note that, up to factors of $-1$, the product of non-monic polynomials cannot be monic in $\Bbb Z[x]$

Balarka Sen

@Daminark Washington is a pretty damn good place

Akiva Weinberger

17:14

(I say "up to factors of $-1$" 'cause you could do it if they things in the product are both negatives of a monic thing)

And every time I try to type "monic", my phone thinks I either meant Monica or mimic.

Eric Silva

Monica polynomials could be a real thing tbh

Daminark

In particular, Washington... DC

Forgot capital letters, how rude

Anyhow

Eric Silva

@Daminark dc sucks man

Akiva Weinberger

Named after someone with the last name Monica? @EricSilva

Balarka Sen

@EricSilva dc is the greatest place on earth

Eric Silva

17:16

lol wut

@Akiva sure, or first name, they sound cool so someone should make them a thing

Akiva Weinberger

Also I'm curious what a mimic polynomial would be

Hm, probably the image of a bijection of some sort

or a fun-named knot invariant

Typhon

@AkivaWeinberger um what?

Fargle

@AkivaWeinberger It looks like a polynomial, but when you try to factor it you're thrown into a turn-based combat sequence.

Typhon

@AkivaWeinberger monic with integer coefficients.

Eric Silva

@Balarka i have a bias against swamp towns

Semiclassical

17:18

okay, this is funny. google "mimic polynomial" with quotation marks

and take a look at the fifth entry :P

Akiva Weinberger

@Typhon Yes, I know.

Semiclassical

(assuming it works the same for everyone else, I guess)

Typhon

reddit.com/r/askscience/comments/4dcie7/… this?

Balarka Sen

What Are You Doing In My Swamp Remix

►

Daminark

Is it really a swamp town? I hadn't realized that

Akiva Weinberger

17:19

The leading coefficient of the product is the product of the leading coefficients, no?

Daminark

KEK @Balarka

Eric Silva

(Swamp here being a metaphor and also because the climate suuucks)

Akiva Weinberger

Leading coefficient meaning the coefficient of the term with the highest power

Eric Silva

@Daminark it's not actually a swamp town

or it might be, but im fairly certain it's not in reality

Daminark

Metaphors are too meta for me

Eric Silva

17:20

the swamp thing was a politics joke

Fargle

Venice: the ultimate swamp town

Eric Silva

but because it has an awful climate people actually believe it sometimes

Semiclassical

i'd make a joke about metanyms but I never remember what they actually are.

Daminark

snaps at self joke

Typhon

I give up

I'm leaving

Daminark

17:21

Oh @Eric I also legit thought it was a physical swamp and was like kek, figures since you're from Florida

Typhon

this conversation is too weird

Eric Silva

@Daminark i also hate actual swamp towns

PVAL-inactive

Similes are like a wind blowing freshly cut grass.

Balarka Sen

heh

Typhon

@AkivaWeinberger Isn't that circular reasoning since factoring implies that I can factor which implies where is some factor(s) which are unneeded?

Eric Silva

17:22

@Daminark a funny thing is that people have talked about draining the swamp and it's gotten even MORE swampy

Daminark

Lol I immediately defaulted to the Shrek joke when I heard that phrase

Semiclassical

A lot of modern politics would be hilarious if it wasn't actually happening.

Eric Silva

honestly actual politics have gotten much more ridiculous than past parodies of politics

Typhon

uuuh

no politics

that's not allowed here

let's talk math

Balarka Sen

Typhon

17:27

@BalarkaSen I hope that a**hole was perma-banned.

Eric Silva

tbf this is like pretty much the way the current administration represents itself to media

PVAL-inactive

If Trump alienates 10,000 of his voters daily how long before he alienates all 62,979,879?

Typhon

and IP banned

Akiva Weinberger

@Typhon How about this. You know Euclid's algorithm for finding GCDs of pairs of polynomials?

Typhon

@PVAL-inactive Never, because one of the 10,000 will end up murdering him.

Semiclassical

17:28

I've linked this before, but it still describes how I feel at this time of the day: youtube.com/watch?v=CahNAauFgys

Typhon

@AkivaWeinberger nope.

Akiva Weinberger

(The division algorithm applied to polynomials, essentially)

Typhon

O.O

Fargle

I think it's really weird that Munkres' proof of the Urysohn metrization theorem uses normality of a regular space with countable basis just to use the Urysohn lemma and then only uses regularity.

Typhon

wait...

polynomials are themselves a euclidean domain?

Daminark

17:28

@PVAL-inactive I don't deal with numbers larger than 7 a la extremely strong Goldbach

Eric Silva

@PVAL like the length of four terms

Akiva Weinberger

@Typhon Yup!

Typhon

dies

Semiclassical

$P(x)=Q(x)(x-a)+R(x)$.

Akiva Weinberger

Using their degree as a measure of size

Typhon

17:29

oh f***

Daminark

@AkivaWeinberger What's the $\sigma$-algebra tho?

Typhon

several of my proofs relied upon them being integrally unclosed

weelp

im done

literally dies

PVAL-inactive

If someone accrues a trillion dollar deficit a year and the interest on the loans given are 5% compounded continuously how long before the deficit out weighs the total amount of currency in existence?

Hint: We are already way past that.

Araske

that sounds like a wolfram-alpha theorem pval

Eric Silva

i mean i feel like there being more debt than money isn't even weird

Semiclassical

17:33

Economics is weird.

PVAL-inactive

Well it is when the debt is entirely the governments.

Eric Silva

i mean debt is a book keeping system and not like an actual thing

Semiclassical

If I actually understood heterodox economics I'd probably want to say something re: MMT.

But I don't and I won't.

Araske

money is numbers

Daminark

@Araske so there is only 7 money

Akiva Weinberger

17:35

That's essentially the idea behind Bitcoin @Araske

Balarka Sen

at least 7, @Daminark.

Araske

there isn't a reason for it to have phisical form

Akiva Weinberger

or ledger systems, I guess

Daminark

@Balarka goldbach

Eric Silva

@akiva this is the essentially the idea behind ALL money lol

Akiva Weinberger

17:36

@EricSilva Yeah but you can hold a dollar in your hand

Araske

its the idea behind fiat currency

a dollar is a piece of cloth and paper

Akiva Weinberger

You can't hold anything in your hand if some ledger somewhere has a "7" next to your name

Eric Silva

@Araske is there really a difference between fiat currency and currency backed by something

Akiva Weinberger

@Araske Hm, fair point

Eric Silva

like is the difference more than superficial i mean

Araske

17:37

fiat currency is backed by something

Eric Silva

sure

Araske

like laws, and government power

Eric Silva

but it's backed by something that a lot of people see as somehow less legitimate than e.g. gold

Akiva Weinberger

Can you replace damaged currency if it's backed by something?

If it's fiat there's nothing wrong with the mint throwing away a ripped $100 bill and printing a new one

Araske

right

thats p much what happens

Akiva Weinberger

17:38

All they lose is the cost of printing one bill, which is pennies probably

PVAL-inactive

less than that certainly

much less than that

Akiva Weinberger

Ledger systems are interesting 'cause they're just a billboard somewhere with people's names and numbers next to them

Eric Silva

it costs like 5 cents

Araske

like you can send damaged currency to the mint and they'll send you back a fresh bill

Semiclassical

federalreserve.gov/faqs/currency_12771.htm

Fargle

17:39

@EricSilva The difference is that the faith-based nature of fiat currency means that it is not backed by a finite resource, and as such isn't subject to massive inflation as that finite resource becomes scarce.

PVAL-inactive

That seems so insanely high.

Akiva Weinberger

and you exchange services for the opportunity to add some value to your number and subtract the same amount from someone else

Maks

Hi ! :)

Eric Silva

@Fargle sure

Araske

@Fargle i mean it still gets tied to commodities but less directly

Akiva Weinberger

17:40

All the complicated stuff with Bitcoin is making sure no one can tamper with the ledger (the billboard)

Daminark

@Araske what if you send it to the coffee beens instead?

Balarka Sen

I want comoney

Akiva Weinberger

I guess it's technically lots of little boards everyone has that are kept in sync rather than one central billboard

Semiclassical

Eh. The element that seems missing in that 'abstract' picture of a ledger system is the extent to which the legitimacy of such a system depends on external power structures.

Fargle

Outside of that, backing by a finite resource is still backing based on faith: one's faith that that resource will remain perceived as valued.

PVAL-inactive

17:41

Even as an individual it is so easy to print relatively nice looking documents at less than 1 ct a page.

Akiva Weinberger

@Semiclassical Bitcoin isn't backed by any bank

It's just backed by the fact that it's hard to tamper with the ledger ('cause of crypto stuffs)

@PVAL-inactive Bills need to be hard to be counterfeited, so they'd cost more, right?

What with the watermarks and stuff

Eric Silva

my point was moreso that perception of things as having value isn't different from the idea behind fiat currency in a meaningful way @Fargle, but the actual currencies "work differently" still, hence your point.

Semiclassical

No, it's not just backed by that. It's also backed by people's acceptance of it as such.

Akiva Weinberger

and some currencies have clear sections in some of the bills

Semiclassical

Maybe 'external power structure' isn't the right way to put it, though.

Araske

17:44

it kinda is?

Akiva Weinberger

You always need people to decide your ledger scribbles are worth more than zero dollars I guess

Araske

like I don't believe anyone is required to accept Bitcoin as valid payment for debts

Semiclassical

Yeah, that's true.

PVAL-inactive

@Akiva I agree but I thought that the fact it's done by much more efficient methods would be more significant.

Semiclassical

It's legitimate in some instances, but not all.

Araske

17:45

its as legit as people agree it is i guess

Eric Silva

tbh wealth is a big spook

Araske

idk, im neither an economist nor an expert on bitcoin tho

Semiclassical

@PVAL-inactive Looking at the attached pdf, I think that the 5 cent mark isn't just the printing costs.

For instance, the following sentence occurs in there: "Variable [printing] costs, by denomination, remained relatively stable between 2016 and 2017 and range from \$ 22.06 per thousand notes for \$1 notes to \$71.37 per thousand notes for \$100 notes. "

PVAL-inactive

Most of the bitcoin community seems to be just speculators.

Semiclassical

So the printing costs would seem to indeed be a fraction of a penny.

Eric Silva

17:48

@Semi sounds like economies of scale

PVAL-inactive

People buying a commodity with no inherent value but with the hope the perceived value increases so they can sell it at a later point.

Semiclassical

I think it's more that it factors in stuff like shipping.

Akiva Weinberger

How do credit cards work?

Not credit

Debit

I feel like it'd also be some sort of ledger-like thing

Faust7

Good morning everyone

Akiva Weinberger

Good morning

PVAL-inactive

17:50

@Akiva do you mean the actual cryptographic operations they do or just where money is coming from?

Akiva Weinberger

Kinda both, they both sound interesting

but the second part seems more relevant

Semiclassical

" The printing budget includes \$418.7 million (62 percent) in fixed costs and
\$255.1 million (38 percent) in variable costs. Fixed costs, which include capital, prepress and engraving, fixed manufacturing overhead and support, research and development, and general and administrative staff, are budgeted to increase nearly \$21.5 million, or 5.4 percent, from 2016 estimated expenses, because the BEP included an additional \$9.3 million for staffing and overtime expenses to support the acceleration of the next design family of notes. "

Eric Silva

@Semi maybe? it seems weird to me that they wouldn't say outright that it included shipping cost

Faust7

do we have some kind of upper bound on what the furthest distance between to primes could conceivably be?

PVAL-inactive

I think the way the chips work is really modern cryptography so reasonably complicated but somehow depends on factorization.

Eric Silva

17:52

but i guess they might

Semiclassical

Looks like I read it a little quick.

The printing budget includes the actual printing costs, but also stuff that's more structural

PVAL-inactive

@Faust7 Do you mean consecutive primes?

Akiva Weinberger

@Faust7 You can have arbitrarily long gaps

Faust7

well yes what the furthest apart two consecutive primes could be

Akiva Weinberger

There standard example is $n!+2$ through $n!+n$

or $n\#+2$ to $n\#+n$ I guess

where, assuming I didn't mess up the notation, $n\#$ is the product of all primes less than $n$

Fargle

17:53

Get those filthy primorials outta my chat

Tobias Kildetoft

@AkivaWeinberger I wonder if it is known whether all even integers occur as the distance between consecutive primes though

Semiclassical

Is there a known estimate of how large in $n$ you'd have to go to achieve a specific prime gap length?

PVAL-inactive

Things like the prime number theorem imply that as you get larger the expected gap between two primes must become arbitrarily large.

Akiva Weinberger

@TobiasKildetoft Hm. Seems kinda Goldbachy?

Semiclassical

I imagine it could be extracted from PNT, yeah.

Faust7

17:54

hmm

intresting

Semiclassical

18

Q: Can every even integer be expressed as the difference of two primes?

Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?

Faust7

you guys are much more knowledgeable then my googling skills

Akiva Weinberger

There's a lot of number-theoretic content that just comes from the fact that $\binom{2n}n$ is an integer, by the way

Semiclassical

Also: mathoverflow.net/questions/111196/…

Akiva Weinberger

Like, you wouldn't expect $2n!$ to always be a multiple of $n!n!$, but it is

and I think a proof of Bertrand's postulate uses that

Semiclassical

17:56

Counting is weird.

Tobias Kildetoft

@Semiclassical Ahh, so what I asked about is open and even open if one weakens it a lot

Semiclassical

Right.

"Plausible but not known" as is typical of these problems.

Tobias Kildetoft

@AkivaWeinberger If you mean $(2n)!$ then that is just an artifact of suitable subgroups of symmetric groups of course

Faust7

yeah that stupid thing with the dots

in a grid

Semiclassical

Well it's also an artifact of $\binom{p}{q}$ is always an integer :P

Faust7

17:58

whats it called

Tobias Kildetoft

@Semiclassical But at least they got fairly close to the twin prime conjecture with their big project (though I think there was some kind of obstruction that prevented it from even having a chance of getting all the way)

Semiclassical

ya, that polymath project.

Faust7

Ferrers graph

Semiclassical

I suppose the simplest proof of $\binom{p}{q}$ (beyond a counting proof) would be to use ...oh, what's the name. Pascal's formula, I think?

Akiva Weinberger

Hm, would you interpret $pq!$ to mean $(pq)!$ or $p(q!)$

What about $p\cdot q!$

Tobias Kildetoft

18:00

Right, they got it down to gaps of 246 according to arxiv.org/abs/1409.8361

Semiclassical

The one that makes the Pascal's triangle work.

Akiva Weinberger

Probably is called Pascal's formula, yeah

Faust7

i never understood pascals triangle very well despite the fact that ive used in several classes

Akiva Weinberger

It's like Fibonacci but 2D and goes down instead of sideways

Faust7

lolk

Semiclassical

18:02

$\binom{p}{q}=\binom{p-1}{q-1}+\binom{p-1}{q}$

PVAL-inactive

Really if you look at it as an induction argument it makes sense.

Akiva Weinberger

Fibonacci's line

Semiclassical

strong induction, I think

Akiva Weinberger

If you tried to make a 1D version of Pascal's triangle that's probably what you'd end up with, tbh

Faust7

is Fibonacci related to e in some way?

Semiclassical

18:04

I don't think so.

Akiva Weinberger

I'm sure somehow

Daminark

Contradiction

Faust7

cause i know when u do pop models e shows up alot in DE's

Semiclassical

I guess it comes down to how natural a relationship one asks for

Faust7

but i rember the fibonacci numbers having something to do with the growth of a bunny population

Akiva Weinberger

18:05

I mean you just need to squeeze $(\frac{n+1}n)^n$ in there somewhere

PVAL-inactive

@Semi I just mean if you look at (x+y)^{n-1}(x+y) it comes down to adding together adjacent coeffs. of (x+y)^{n-1}

Akiva Weinberger

In any case $1\cdot2\dotsb n$ is a factor of $(n+1)\cdot(n+2)\dotsb2n$

so you can do careful counting of the prime factors of each of those

and then, I dunno, Erdős magic happens

Faust7

lol

thats a very big way of doing things

Waiting

18:20

@Hippalectryon o/

eurocoder

18:33

What is meant by a 'real normed vector space'? It seems that $\mathbb{R}$ with the absolute value function fits the description..but are there other spaces that are also called real normed vector spaces? Could $\mathbb{R}^3$ or $C^1([0,1])$ also be called real normed vector spaces?

Fargle

@eurocoder I'd need to check for $C^1([0,1])$, but $\Bbb R^3$ is definitely a real normed vector space--it is over the field $\Bbb R$, and it has as a norm the usual Euclidean distance from the origin.

Tobias Kildetoft

@eurocoder Any real vector space with a norm fits the bill

Fargle

Okay, yeah, $C^1([0,1])$ works, with the norm being $\sup |f(x)| + \sup |f'(x)|$.

And it is also over $\Bbb R$, as is more or less clear.

Tobias Kildetoft

@Fargle Why add sup of the derivative?

Fargle

@TobiasKildetoft I guess you don't have to, but adding the sup of the derivative makes it a Banach space.

Tobias Kildetoft

18:45

I thought it already was one

Eric Silva

uniform limits of $C^{1}$ things can be differentiable nowhere

Fargle

^

Tobias Kildetoft

Ahh

Hmm, so we need to work over the complex numbers to make this a $C^*$-algebra, right?

(and take all continuous functions rather than the $C^1$ ones)

Eric Silva

isn't being over the complex numbers part of the definition of being a $C^{*}$-algebra

Tobias Kildetoft

@EricSilva Yeah

Eric Silva

18:50

mmk

gtg give a lecture

bye chat

Tobias Kildetoft

Whenever I get an upvote on a question or answer and I have not asked or answered anything in a while, I know it will be one of a fairly small set of such that got it.

eurocoder

Ok thanks guys!

So it seems that just about everything is a 'real normed vector space', and in the case of real normed vector spaces, boundedness is equivalent to continuity and hence the strong and weak topologies coincide. So when do they not coincide?! It seems like it would be really unusual for them not to coincide?

Tobias Kildetoft

@eurocoder What about the vectorspace of all maps from the reals to themselves?

Or probably just the space of continuous maps on an open interval

Richard

Does Little Fermat imply that in $\Bbb Z/p[x] $ one has $x^p=x$ if $p $ is prime?

eurocoder

@TobiasKildetoft Hmmm..well I asked about $C^1([0,1])$ above and it was confirmed this is a real normed vector space, so I assume $C^0((a,b))$ is also a real normed vector space, and hence boundedness = continuity, and the topologies coincide?

Tobias Kildetoft

19:04

@eurocoder Note that the norm defined used the fact that the domain was compact.

for the open interval, just consider something like $1/x$

eurocoder

@TobiasKildetoft I really don't see how that fact changes anything..it seems $C^1((a,b))$ fits all the requirements of a real normed vector space? And being such a space is the sole requirement for boundedness to equal continuity.

Tobias Kildetoft

@eurocoder What norm would you put on it? And my example clearly shows that there are non-bounded continuous functions on an open interval

Ohh, I misunderstood what you meant with bounded iff continuous. Anyway, you can certainly put a norm on it, since it is isomorphic to the previous example as a vectorspace. It is just not obvious what that isomorphism would look like

eurocoder

@TobiasKildetoft I would not put any explicit norm on it, it holds for all norms as far as I know. I don't think your example of f(x) = 1/x holds as it not linear?

Tobias Kildetoft

@eurocoder Yeah, I was thinking of the functions in the space itself, rather than functions on the space

eurocoder

@TobiasKildetoft I really am just trying to understand what sets are 'left out' if we use the weak topology instead of the strong topology. I just can't see how they contain different sets!

Tobias Kildetoft

19:16

But it is not clear to me that a norm always exists if the dimension becomes large enough (though it might do so by some argument I am missing)

@eurocoder I am not even sure which ones those are. Choosing a topology is of course much weaker than choosing a norm

eurocoder

@TobiasKildetoft Well my situation is, I am taking X to be a real normed vector space. And then I consider the norm (also called strong) topoolgy, and the weak topology. And supposedly the weak topology may by smaller than the norm topology, so I was wondering what sets can be discarded. I think I'll just move on and maybe I'l have better idea as I cover more material!

Tobias Kildetoft

@eurocoder Not sure what you mean by discarding sets. You mean which are open in one but not in the other?

Daminark

@eurocoder I imagine this is related to the question you made about continuity with respect to one topology but not the other, right?

So, if you're in a finite dimensional space, I think weak and strong topologies are the same

In an infinite dimensional space, you tend to lose quite a lot. No bounded sets are open, I think (and the topology ceases to be metrizable)

PVAL-inactive

19:44

A real normed vector space has two pieces of data. A vector space $V$ and a norm $n: V \to \Bbb R$. It doesn't make any sense to ask whether a particular vector space is a normed vector space.

You could ask that it admit the structure of a normed vector space but that is a different question.

Daminark

Whoa @PVAL your picture changed

PVAL-inactive

My ip changed

skullpatrol

Does that mean you're active now too?

Daminark

Ah, iseewhatyoumean.jpg

PVAL-inactive

I actually see a different picture next to my name than the one next to the chat box.

Balarka Sen

19:55

A few years ago all the system generated gravatars changed for some reason

PVAL's this avatar is the changed one

for some reason he's got two avatars now

PVAL-inactive

I always thought it was based on ip.

skullpatrol

Me too

Tobias Kildetoft

@PVAL-inactive Right, but IP when you sign up

Balarka Sen

If I get on a different machine I don't get the avatar change

PVAL-inactive

well mine's changed multiple times.

Tobias Kildetoft

19:56

It is not meant to change based on where you access from

Daminark

So it seems like they have wears sunglasses an IP system

Balarka Sen

@Daminark ayy

PVAL-inactive

Mine was originally some color, and when I moved it changed to burnt orange (which made sense).

skullpatrol

What if you use a VPN?

Daminark

@skullpatrol then gg no re

Transcript for

$Mathematics$ Mathematics