Mathematics - 2017-06-27 (page 1 of 6)

lattice

00:00

If I localize a ring at just the neutral element, I get the same ring again, right?

Fargle

00:11

Wait, by neutral element, do you mean the additive or multiplicative identity? Just to be clear.

lattice

Sorry, I mean the multiplicative identity

for the additive identity we get the zero ring

Fargle

Right, right.

I should think so, actually.

lattice

okay, thanks!

At least it's something isomorphic to it I think

I sometimes don't really know if I should say "it's the same" or "it's isomorphic"

Fargle

Checking the definition, you want $(m,1)$ and $(n,1)$ to be equivalent if $1(1n - 1m) = 0$, or $m = n$, so you're creating a ring isomorphic to $R$ whose elements could be written as $\frac{r}{1}$.

Unless you're still working with the exact same objects, saying "it's isomorphic" is more precise.

lattice

Okay

Fargle

00:16

In the case of localization, you're constructing a new thing--namely, a ring structure on $R \times S$, so it's different, but isomorphic.

lattice

Yeah

I think my Professor is not so strict with the notation etc.

Fargle

In that case, you'd be fine either way. After all, two isomorphic objects are basically the same thing, up to relabeling, so conceptually you wouldn't be wrong.

Just that there's a pedantic way to look at it.

lattice

Yeah

and I guess if we kept track of the correct notation all the time, then algebraic geometry wouldn't be much fun

Fargle

That I can't speak to as much--I don't have the right background for algebraic geometry yet. Though I really want it!

lattice

Haha me too

it's a tough way though :D

Adeek

00:23

man I have a confrontation today with my supervisor

Leaky Nun

00:42

why?

Adeek

He was always rude to me so I changed to someone else

Leaky Nun

you mean you had a confrontation?

Adeek

yeah

He was being super rude to me and not being friendly

so I wanted to see if I could change him and I did end up working with someone else

Leaky Nun

lol, the tense conveys meanings other than time

if you have a confrontation, it sounds like you planned it

Adeek

yeah i guess

Daminark

00:54

Damn, sorry

Hope your new adviser is better

Adeek

me 2 I know one guy who I want to work with, so I hope he accepts me.

Daminark

Oh, wait so you don't yet know if you'll get that new adviser? What happens then?

Adeek

@Daminark my old advisor was very rude to me and if I made a small mistake he would like throw a fit and wouldn't explain to me what is wrong or anything like that.

@Daminark I will see what will happen. I just don't think about it until this potential advisor emails back.

Daminark

Alright, good luck

Adeek

thanks

Daminark

00:59

Which year are you in grad school now?

Or I guess, which did you just finish?

Adeek

@Daminark yeah I am in grad school atm

Daminark

No I mean, are you a first year student, second year, what?

Adeek

first year @Daminark

Twink

wtf is $\sigma$?

Daminark

01:14

I see

Leaky Nun

@Twink a function

Adeek

anyway I am happy I changed now I wouldn't be able to take it to work with him @Daminark

Akiva Weinberger

01:27

1

reddit.com/r/tifu/comments/2livoo/…

2

reddit.com/r/AskReddit/comments/6jix5j/…

.

(Remind me to send the first link to my English teacher when I wake up.)

Daminark

Yeah for sure

Send the first link to your English teacher when you wake up

Semiclassical

01:48

@Twink If you like, $\sigma(w)\equiv \dfrac{f(z+w)-f(z)}{w}-f'(z)$

which goes to $f'(z)-f'(z)=0$ as $z\to 0$.

why they only seem to think of it as a function of $w$...hrm.

I can't quite make sense of what they're doing, alas.

blah, should have been "as $w\to 0$" above

Avantgarde

@Twink It's the terms (to be multiplied by $w$ later) in the Taylor expansion of $f(z+w)$, excluding the first 2 terms (one of which cancels $f(z)$ and the other is just $f'(z)w$

wait

I'd need to see this

why are they even doing this. I've not seen such a derivation of Cauchy Riemann equations before

Semiclassical

02:34

If I had to guess, they're just suppressing the $z$-dependence for the purposes of their argument.

Zee

04:05

Hello

user84215

04:24

For a point of a regular surface with positive Gaussian curvature, if it is a local maximum for one of the principal curvature and a local minimum for the other one, why are these principal curvatures equal to each other ?

ಠ_ಠ

Any topologists around who could help me with this? math.stackexchange.com/q/2336320/169780

Daminark

Hey everyone!

Zee

@Daminark sup dank

Daminark

How's it going?

Zee

04:41

Alright, drinking beer and eating mac

You?

Btw , it's possible to derive all of algebra from PDEs

Theoretically at least

Daminark

That sounds suspicious

And I'm doing alright, thanks

Zee

04:59

@Daminark well , you can simulate Newtonian mechanics with PDEs, so all we have to do is simulate the body of grothendieck, and then we can have all he produces, we can even do better and simulate the perfect algebraist

Iff such an algebraist exists

The essence is that you can derive ANYTHING from PDEs

Daminark

Lol, free will tho

Zee

What about it?

Daminark

You can't predict the actual results of algebra

Zee

Well I see two arguments here, are they the same?

am not sure what you mean by free will but for predicting algebra, all I have to do is simulate grothendieck and run different variations of him

or just simulate grothendieck and feed him different problems

Daminark

You can't run a simulation of Grothendieck is the problem

Zee

05:10

Why not?

Daminark

Like, you can't replicate a human being because the actions aren't determined by Newtonian mechanics

Zee

Lol

Come on

What else? Magic?

Daminark

Thought

user84215

For a point of a regular surface with positive Gaussian curvature, if it is a local maximum for one of the principal curvature and a local minimum for the other one, why are these principal curvatures equal to each other ?

Zee

@Daminark that vague

Daminark

05:19

Well, for what it's worth I'm leaning more to pluralist than physicalist

Zee

You took this from the realm of science and math to philosophy, where things are based on intuition and unsupported by either reason or experiments

Daminark

I mean, you can't prove via experiment whether pluralism or physicalism works

Zee

so why adopt any of those?

We know that there is science and it obviously works

Why not just live within that frame work?

Daminark

But if our thoughts were determined by initial conditions at the big bang, then that would undermine their validity, including the thought that this was the case

Zee

Why would you say this??

How does that undermines our thinking?

Daminark

05:27

Because then, that which we come to believe cannot be said to derive from reason, it was determined by some initial random conditions at the beginning of the universe that are not necessarily bound to lead people in the right direction

Perhaps we were determined to think that truth values acted in a certain way when they really did not, crippling reason itself. Anyway, beyond all that speculation, there's really not much to say if that were the case, everything halts

Abcd

Daminark

And when you were mentioning that we know science, so why adopt the anything in the physicalist vs pluralist debate, making the claim that one could find a few PDEs floating around and simulate Grothendieck perfectly is necessarily adopting a physicalist position

Abcd

Can someone help me with this question? It's difficult.

Zee

Look up the questions, am sure there is dozens of similar ones to it

Abcd

@Zee nope

Zee

05:31

Try the physics chat

Abcd

Ok

Zee

@Daminark I don't see how other philosophical positions can bring back free will, aside from some form of supernatural world

Daminark

Additionally, note that even if you could conduct this simulation, that doesn't mean that algebra derives from PDE

Okay perhaps free will is an iffy way of putting it, let's say ability to reason

Whether those two are the same, maybe, haven't thought about that much yet

Zee

Well any result of algebra that has been discovered can be simulated , and any result that could have been discovered by a human as well

Daminark

And I mean, if supernatural means aphysical, then I mean sure.

Typhon

05:35

@Abcd no units, no answer...

id pick e

@Zee its a math problem...

Zee

No, it's not that , super natural means some kind of magical thing that we can't explain, aka black box, but a black box can explain anything so it's not really fair to use it

Daminark

And the result itself is not simulated or contained, merely its derivation by someone is, in retrospect. Epistemologically those two are different

Abcd

@Typhon Assume SI units...

Typhon

@Abcd that's still unsolvable.

how do i know the answer is in seconds, milliseconds, minutes, or hours?

Daminark

I mean, so we don't know if our sense perception is even correct, and that our subjective experience has any truth value to it at all

Abcd

05:38

@Typhon All that has to be assumed

Typhon

...no

Daminark

To engage in thought whatsoever, things need to be black boxed on basis of plausibility

Typhon

1 hour = 60 minutes

for instance

Abcd

@Typhon See this: clay6.com/qa/52505/…

Daminark

One of those things which is plausible is that we have the ability to reason about things, that's no more of a black box than science, and perhaps maybe even less

Abcd

05:38

But I can't understand that...

Typhon

so the answer could be in a different unit and therefore differ by a multiplicative constant

Abcd

@Typhon Much ado about nothing

Typhon

@Abcd wat. 60 minutes is an hour.

Abcd

@Typhon Did you see the solution I linked you to?

Typhon

thats not relevant

Daminark

05:41

So then given our apparent ability to reason, we ask whether we know it works out properly if we are to suggest that thought is determined by the laws of physics versus random or based on some actual ability humans have, which may derive from a non-physical reality, to reason. The answer is that we don't know. And neither claim is on a metaphysically stronger basis than the other

Typhon

@Abcd no. it cannot be answered properly. all of those answers could be either right or wrong because they are not measurements.

suppose it is hours

Zee

Testing

Typhon

and you answered for minutes

Abcd

ASSUME SI units @Typhon

Typhon

@Abcd hours and minutes are si units dumbass.

Zee

05:43

@Daminark you bring many points, I can't really address all of them at once...

Abcd

@Typhon second is SI unit for Time

Typhon

SI units are not single unit

Daminark

Hence, we do not really have particularly good reason to believe that PDE contains algebra in the way you say, and again, that's epistemologically distinct from derivation, the way that you can derive the implicit function theorem from the inverse function theorem

Lolol, fair, I tend to write this stuff in a very train of thought style

Zee

I can even do better than my original argument

Daminark

Let's hear it

Typhon

05:44

@Abcd that is beyond narrow minded. so is hours, minutes, and days.

Abcd

@Typhon They aren't.

International System of Units

The International System of Units (French: Système international d'unités pronounced: [sistɛm ɛ̃tɛʁnasjɔnal dynite]; abbreviated as SI) is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units. The system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as a result of an initiative that began in 1948. It is based on the metre-kilogram-second system of...

Have a look at the table.

Zee

Let's say you wanna solve some kind of algebraic symbolic relation

Abcd

@Typhon ASSUME - m/s, s, m . I hope that's better.

Zee

You just simulate the symbols on the complex plane and simulate all possible movements of those symbols that correspond to the algebraic rules, and then just see if any PDE would give you such a result

Daminark

I mean this depends strictly on choice of notation, and again is epistemologically different from derivation

A priori you have no idea which PDE would work, even with notation given

Zee

05:47

sure but you can standardize the notation

Daminark

The only way you could know is by already knowing the algebraic results

Which defeats the purpose of this

Zee

Well you have both sides of the equation, you wanna prove it

So you keep moving the left side in a way until it gives the same image as the right side

Typhon

@Abcd you missed my point altogether. the thing you linked is a trick question. In true physics, the lack of a unit means just a number, so all four answers are incorrect. Even if they are some mystery unit assuming seconds is without justification and incredibly narrow-minded. SI is a system of units, and not just one unit for every measurement.

Zee

Then you take such a PDE, and "slow it down" and see how it maps the symbols at each T

Typhon

so... you're still left with seeing which answer is if the form k*x = y where y is your solution and k is some constant. That's just silly.

is this college physics?

i sure hope not

Daminark

05:49

If you're just homotoping the symbols directly then you're not even generating algebraic results. If you're doing it under rules of manipulation, then again, you fail to have information about algebraic results from the PDE because verification involves testing it against the algebra, which you have to know already

Zee

It's about finding the proof

You have the result you want to find out how to manipulate the symbols to get the proof

Abcd

@Typhon No High School (Grade 11) Physics.'

Daminark

Sure, and what would the process be to do this?

Zee

Formalize the symbols as a figure on the complex plane

Typhon

mumbles curses about Obama and educational budget cuts

Zee

05:51

And then formalize the symbolic manipulations as geometric mappings on the plane

Daminark

Which symbolic manipulations?

Zee

What ever your working with, if I wanna find a fact about a polynomial, it would have to follow the algebraic rules for manipulating polynomials

By fact , I mean a formula

You gotta stop being so skeptical of the world

Daminark

Alright, so using my knowledge of PDE and the fact that I don't know any algebra (let's say we're trying to derive the very first result), how shall I figure out whether a given symbolic manipulation is legal or not?

Typhon

@Abcd alrighty. let me correct a few things, because I know a bit more and you are mistaken, kid. First of all SI units refers to the usage of things like using meters and centimeters over inches and feet. It doesnt mean assume it is in meters. In fact assuming units is always wrong to do unless it is a very general identity in which case that is ok. Secondly, units are always required. At the higher math level they are elements of a abstract number rings which means...

Zee

idk at that point...

I was teasing you with PDEs you know...

Typhon

05:56

that their formulae and manipulations only apply to the units and NOT to the numbers. It's like forgetting i with imaginary numbers. You technically end up doing it wrong bug time.

Daminark

Oh I know but I'm just bored so I actually wanted to pursue this

Typhon

So... um.

Daminark

It's fun

Zee

I still don't see the problem with simulating grothendieck...

Typhon

please keep in mind that 11th grade is primarily teaching you horrible physics. XD

@Abcd anyways good luck.

Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions. The conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in...

Daminark

05:58

I mean, basically that you don't really know that this should work, and if you did know

Zee

Wait, you can try different axioms, and then simulate them to see which ones give you the desired algebraic relations you want and don't want in a system

Daminark

You'd still have the problem that you're using the fact that someone has already derived the results, and also knowledge of physics, to pull algebra out of PDE

Leaky Nun

@Abcd second is SI base unit, hour is SI derived unit, both are SI units

Typhon

^^

Abcd

@Typhon Ok. Thanks.

Daminark

05:59

This is different than using reason to derive formal results from other formal results

Abcd

@LeakyNun Oh. I intended to say: SI base units.

Zee

@Daminark you gotta be more creative, I would just simulate myself now, and then see what algebra would the PDE give me

Typhon

@LeakyNun they claim that if it isnt listed on the wiki Page for "SI", then it isnt a unit in SI.

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$Mathematics$ Mathematics