2:57 AM
hey yall

3 hours later…
5:43 AM
@LeakyNun, Is this correct argument to show that if $v$ and $w$ are eigenvectors with different eigenvalues, then they are linearly independent: In $c_1v+c_2w=0$, if $c_1=0$ then $c_2$ has to be zero, since $w$ eigenvector hence nonzero, and if $c_1,c_2$ both nonzero then $v=\frac{-c_2}{c_1}w$, so $w$ is in eigenspace of $v$.

6:00 AM
or just multiply the equation by λ1 lol

@LeakyNun, when we talk about different eigenvalues e.g., here, we mean that they are not multiples of each other, right?

@Silent we mean they are different
(the eigenvalues are elements of the scalar field, not vectors)

@LeakyNun,what does $m_T$ mean here?

6:21 AM
How to integrate: $$\int k x^2 e^{-ax^3}dx$$
k and a are constants.
@LeakyNun Could you give it a look please?

u=x^3
@Silent minimum polynomial

@LeakyNun Wow, thanks!

@LeakyNun thanks!

lol, 5 characters helped the first one and two words helped the second one
2

@LeakyNun Which means it took you more words to snark about the help you gave than to actually give said help :)
2
(of course, this last message was longer than all of those combined...so yeah)

6:42 AM
I am currently thinking about continua, but nothing yet
Besides crazy things, I also like nice things such as continua, very round things and continuity

I have a function: $f(r)= \dfrac{\rho_o}{\epsilon_o r^2 3a} (1- e^{-ar^3})$
How do I investigate it at $ar^3 <<1$

if $e^{-x}$ is small, you can expand it as a taylor series and then 3rd or higher order terms can be truncated. Not sure how mathematicians does the analysis though, they probably use big O notation somewhere

its from physics and we havent studied taylor series in maths yet.

then learn it
in research you don't go like "oh i haven't learnt this idea so I just throw this problem away"
you go like "oh I haven't learnt this idea lemme grab a book and learn it"

@LeakyNun But I am going step by step... Functions almost finished then next chapter is limits
why to jump

6:47 AM
well, a complication of school is that you are not allowed to write out of syllabus answers, so I don't know what else can do it besids taylor

nvm

@Abcd what have your maths component of your course covered so far?

@Secret Functions, functional equations, graphs, transformation of graphs, properties of functions.
Besides previous year's chapters like full algebra, analytic geometry, trigo etc
Okay, I'll do this problem after completing the necessary portion in maths.

Hmm...
$$f(r)= \dfrac{\rho_o r}{\epsilon_o r^3 3a} (1- e^{-ar^3}) >> \dfrac{\rho_o r}{\epsilon_o 3} (1- e^{-1})$$
other than that, I don't know what else we can say if we cannot use taylor series
wait a minute, mistake
$ar^3 << 1 \implies -ar^3 >> -1$
$$\dfrac{\rho_o r}{\epsilon_o r^3 3a} (1- e^{-ar^3}) >> \dfrac{\rho_o r}{\epsilon_o 3} (1- e^{-ar^3}) << \dfrac{\rho_o r}{\epsilon_o 3} (1- e^{-1})$$
argh, I hate whenever the inequality signs are opposing each other, because nothing useful can be concluded from that

7:50 AM
Episode 2 of The Lighthouse

2 hours later…
9:22 AM
Is it true that for a $n \times n$ matrix $A$ $\left\lVert Ax \right\rVert \gt 0$ for all $x \neq 0$ if and only if $A$ has rank $n$?

@philmcole nope. consider $-I$.

Umm
$-I$ has full rank and it's true that $\|-x\| > 0$ for $x\ne 0$
This statement is true, $\|Ax\| > 0 \iff Ax \ne 0$, and if that holds for all $x\ne 0$ it means the kernel is trivial. So by rank-nullity, that's the case iff the rank is $n$

oh what am I saying

Cool thanks @Daminark

No problem @philmcole, and lol @Leaky we all have brain farts sometimes. Maybe the positivity business threw you off for a sec

9:57 AM
God**************, give me a 100%! :

10:08 AM
@LeakyNun, is this kind of approach valid, even when two or more eigenvalues are equal?

We are happy to announced that after so many repeats, this game is finally and utterly, brutally beatened

Which game is this?

2 hours later…
12:10 PM
@Silent as long as the eigenvectors are linearly independent

2 hours later…
2:19 PM
@TedShifrin Hi Ted,

I believe in the solution to 9.4.18c in Multivariable Mathematics there might be a mistake. It says the orthogonal matrix is

$$\frac{1}{5} \begin{bmatrix}3&-4\\4&3 \end{bmatrix}$$

and $\mathbf{y}=Q^T\mathbf{x}$ should give $y_1 = 5 y_2^2 - 1$. However this doesn't feel right, since this would be an upwards parabola in the $y_1$ direction. I think it should be a downwards parabola in the $y_2$ direction.

As eigenvalues I got $\lambda_1=25$ and $\lambda_2=0$ with eigenvectors $v_1=\begin{bmatrix} 4\\3\end{bmatrix}$ and $v_2=\begin{bmatrix} -3\\4\end{bmatrix}$ which giv

2:55 PM
@LeakyNun thank you!

1 hour later…
4:06 PM
any help to understand the proof of -
0

Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that $$f: \mathfrak{g} \rightarrow T_eG \\ X \mapsto X(e)$$ is a linear isomorphism. The only thing that I have left to prove is surjectivity...

1 hour later…
5:13 PM
How to create an infinite series that grows without bound:

^?

Take a stork, trade it in some financial system, then recursively dress it with financial products after products
then with a nonzero probability, the price of said stork will be inflated without bound
and this is why the market constantly crash (I think)
(too much speculation)

lol, why the protest all the sudden

Well, I was reading some energy policies of the energy campaign of all australian parties to analyse for something. Currently reading the last party which is on the futhest left
(Perhaps I will found myself talking nonsense later) Currently, I think the complex layering of financial products on top of each other makes it very hard to track how the value correlates to the good it based on, thus allowing the value to inflate without bound as for each nesting, the value shifts further from reality due to the statistics involved in the calculation
so when the value is gone so far from reality that no real goods can correspond to it, suddenly there is huge debt and the market crashes

thats not entirely accurate
if you dont wish to participate in derivative markets, you dont have to.

5:22 PM
@philmcole I agree something is amiss there. But note that if you change an eigenvector by a sign, the "direction" of the associated axis in the new coordinates also changes. But at any rate, that matrix is messed up. Sorry.

what do you think is the main reason that markets keep crashing?

but they provide value. Say the COO of American Airlines doesn't want to speculate on the price of oil in 6 months time. She'll cover herself with a future that fixes the price for 6 months. Thereby avoiding speculation, and having (more) concrete financial reports.
I haven't seen a market crash since 2008. Currently the political turmoil. Although I couldn't compare it to 2008. nowhere near it.
2008 isn't speculation. It's plain fraud. Moody's and S&P rated garbage collateralized debts as class A. And got away with it.

@JoeShmo ??????????? There have been crashes since 08

of the same magnitude?
@EricSilva

There havent been any quite as bad globally but there have been multiple crashes on similar orders of magnitude
Especially outside the us

5:33 PM
are they getting more frequent or more or less the same rate?

@philmcole Oh, never mind. It's correct as is. $(3,-4)$ is an eigenvector for eigenvalue $0$ and $(4,3)$ is an eigenvector for eigenvalue for $25$. It is correct. You've chosen a different order and different eigenvectors.
Heya @JoeShmo and @EricSilva.

the Greek debt crisis, and the Chinese market crash like 3 years ago were pretty bad
@Secret small crashes happen literally all the time and always have, big crashes like 08 happen once every couple decades and probably won't ever stop happening, maybe they'll get more frequent, maybe not, idk
Recovery time is better now but who's to say that will keep up
@TedShifrin sup

Been hosting my sister & bro-in-law ... now down to just one for the weekend. They depart at 5:30 AM Monday morning. You recovered from your ordeal, Eric?

I am not very literate on finances thus I am not sure if people have some consensus on what is the underlying causes or leads to the crashes

5:38 PM
Which ordeal, there have been many

I might check later...

the Greek debt crisis wasn't triggered by a failure of the market. the chinese bubble bursting is arugably one. although they recovered in 6 months time. again, not to be compared with 08

Oh, more than I knew? Sorry :( ... I meant the GRE, of course.

@JoeShmo that's similar in order of magnitude to the 08 one so idk what u mean

sup @TedShifrin homework saturdays.

5:41 PM
@Secret I mean crashes are just part of how capitalism works but they happen for loads of different reasons

Speaking about capitalism, I still have no idea what could possibly replace profit as an economic incentive

forced labor

profit and human greed seemed to me to be one of the primary reason why the free market keeps going wrong

JoeShmo: Not for me :) ... Although I do have my hour "class" on calculus via document cam (working with twin kids of old friends).

that's awesome. Are you prepping them up to take the putnam exam?

5:44 PM
@Secret when people theorize along the lines of "what would a post-capital economy look like" they usually just spout nonsense. It's like trying to think about what happens post singularity

well, singularity is a plausible post-capitalist economy. or we're all dead. either way, its post-capitalism

so we only have one known solution to the free market problem, socialism?

@JoeShmo the singularity isn't an economy so idk what you mean

if you have robots working for free, with no moral repercussions, you sieze to have a constraint-based economy

@Secret socialism isn't a "solution" to the problem of world capitalism, it's the thing that necessarily follows it's breakdown (if you believe marx and the dialectic, if you don't it's up for debate because marx was wrong about a lot of shit)
@JoeShmo part of the idea of the singularity is that runaway tech development would be unimaginable, so idk what you mean, the whole point is "we don't know what happens post-singularity"

5:49 PM
Ok then I have absolutely no idea what can fix the economy

that was just a jumble of words.

@JoeShmo that will be ideal, but somehow wealth need to be prevent to be concentrated to a small group of people, and I don't see any effective solutions

@Secret neither does anyone else i guess
@JoeShmo what was

your last paragraph. i dont see how its relevant
or what it means

@JoeShmo you're somehow equating "singularity" with "robots doing all our labor for free" but that's not what the singularity is

5:51 PM
@Secret musk is talking about universal income. democratizing wealth, and so on. its a really good question. we don't have models for this economy by nature of the fact that it never existed
you're equating. I'm stating implication. Well, that's one way it could turn out, anyway.
singularity => free robot labor

it's not an implication

says you

the point of the singularity is that we don't know any implications of it
that's why it's called the singularity

speaking of singularity, the MIT summer school organizer has a new paper on quantitative backwards estimates for Hawking singularities
in dimensions greater than 38, which is a bit funny

i think its reasonable to interpret singularity as automation of human-level (and perhaps beyond) ability to reason and call forth creativity.
if we manage to tame it, we have a wonderful workforce on our hands
one concern is the moral concern. If these entities are able to reason, perhaps feel emotion, is it even right to do something like that?

5:56 PM
For me, I will use them until they start feeling emotions, then I will treat them as equals
Or in order to keep them as machines, actively wrote failsafes so they cannot achieve sentience
it's a hard problem in general, because you make them because you want them to do all your work, but then once they achieve sentience, they are essentially our slaves thus violating moral codes

@JoeShmo i think i misunderstood you, yeah sure it's one possible outcome, maybe not an unreasonable one, but i still think this kind of speculation is on some level pure gibberish, which is the point of my comment

@0celo7 speaking of GR, i have some gossip ive heard but i should not say it in a public chat

6:20 PM
@EricSilva hi, why not

Bc it was professors being salty about other professors

@Secret inequality is a factor. the current conception is not quite accurate as merely an imbalance. it to some degree represents exploitation of the 99% by the 1%. in sense, marx was right, its just (almost) nobody realizes it yet. yes there is some consensus about "how to fix it" but alas it tends to split along left/ right lines.
@EricSilva are you looking into GR? seems like it would be harmless. what school are you going to? math major?

I'm at uchicago and I'm interested in learning abt mathematical GR (idk any yet) and more broadly interested in geometric analysis (which ik a bit about)

@EricSilva so are you math or physics major?

Math

6:25 PM
since you wont mention the rumor, heres something else for your entertainment :)
in theory salon, Aug 13 '16 at 23:01, by vzn
re Tenev/ Horstemeyer https://arxiv.org/abs/1603.07655
@Secret thats a strange ideology that almost nobody else is espousing, quite the opposite sentiment is mainstream (using robots as slaves but not with that terminology, ie more servants/ labor). but honestly did imagine ~2½ decades ago that a "robot rights" movement might arise somewhat analogous to "animal rights". (PETA)

I have a tendency to came up with ideologies that sounds very unhuman, because my affinity to the unhuman is my nature

7:00 PM
guys, suppose f: R^n -> R is smooth.
Is ∂_i(f(c*x)) = c^n * ∂_i(f(x)) ?
no that sounds like nonesense

7:32 PM
shit. does anybody understand diff geo?

7:43 PM
@JoeShmo yes, what do you need
you want the chain rule, but need to be careful with your parentheses
$D(f(cx))=c (Df)(cx)$

the exercise asks if to show that the volume of the n-sphere with radius R is R^n * volume of the unit n-sphere
fair enough, intuitive enough. then it says use the fact that the volume form on the sphere is v = n(x) . *dx
and then see what v(Rx) is equal to. But v(Rx) = v(x)

what is n(x)
and dx
what?

i probably get the R^n factor when i pullback to \bR^n
too many R's.
n(x) is the unit normal
*dx is the hodge operator applied to the vector dx
when i pullback to R^n, i bet the determinant will give me the factor im looking for. but im still fuzzy

I don't have a good non-measure theoretic proof for this off the top of my head
If you can use some measure theory it follows from the easy scaling properties of Hausdorff measure

whats the measure-theoretic proof

7:51 PM
Ah, that's a circular proof

can't use measure theory probs. but i bet the scaling properties of the hausdorff measure (Whatever that is, i did encounter it on wiki while researching this question) have the same principles baked into it
i have to pullback the n-sphere to [0,2pi]^n
and im integrating over the same form
except when the sphere has radius r, im stretching each direction by a factor of r.

Ok, there's a much better proof of this
So, abstractly, the 1-sphere and the R-sphere are the same manifold
But they have different metrics

yuh
the matrix is r * I

the only difference is the overall scale

who's determinant is r^n ... :) ?

7:54 PM
so you have $$vol(S_1)=\int \omega_1$$
and $$vol (S_R)=\int \omega_R$$
now find the $k$ such that $\omega_1=k\omega_R$ and you can figure out the rest
and that follows from knowing how $g_1$ and $g_R$ are related, which is what you said

what is omega_R?

the volume form of $S_R^n$

the volume forms of S_R^n and S_1^n are the same
as far as i can tell

why would that be if they have different metrics

what do you mean by different metrics

7:57 PM
what does one usually mean by things being different

the volume form is given by v = n(x) . *dx
n being the unit normal, again.
so v(rx) = v(x)

the volume form is given locally by $\sqrt{\det g}dx^1\wedge\cdots\wedge dx^n$

g being the embedding of [0, 2pi]^n into the n-sphere?

$g$ being the metric on the manifold...

you lost me :)

8:00 PM
you've lost me too

i hope we find each other

you asked me a Riemannian geometry question, but don't know what a metric is?

um, no
i have no idea what i asked
look, im following broken notes, translated from french to english
and im trying to reverse engineer everything that he is saying in a broken fashion. and its been miserable

why are you doing that
there's plenty of good books out there

yeah, but i only picked up a good book last week, and its the end of the semester
so its too late for thata
either way, he insists that the volume form is mu = v = n(x) . *dx
these are the official notes of the class btw, i didn't just pull them out of thin air. the instructor is french, so he (thinks he) understands them. and he forced them down our throats
@0celo7 ok yeah thats what im alluding to. g_i is the local representative of the atlas g? where the atlas induces a metric?

8:10 PM
@JoeShmo I have no idea what you're talking about, sorry
I am talking about the usual way one defines a volume form locally

ok

If $50x^2+51x$ doesn't divide by 13 does X have any special properties?

8:31 PM
You mean, other than being NOT congruent to 0 or 6 mod 13?

8:55 PM
Is this correct $P(-.3960<z<.3680)=P(z<.3680)-P(z>-.3960)$

2 hours later…
10:42 PM
anyone here?
anyone here is good with logics?

i can try

10:57 PM
Hello!!

I want to show that $\|x\|_{\infty}\leq \|x\|_q$.

Do we show that as follows?

$$\|x\|_{\infty}=\max_i|x_i|=\left (\max_i|x_i|^q\right )^{\frac{1}{q}}\leq \left (\sum_{i=1}^n|x_i|^q\right )^{\frac{1}{q}}=\|x\|_q$$

Is everything correct? Could I improve something? Or is this not a complete proof?

11:14 PM
That seems right. @Mary

@Fargle!!

@Ted!! What's going on?

Not too much. Just finished my document cam FaceTime lesson with two kids.
heya @Alessandro

Sounds like fun.

11:18 PM
Good exercise for you, Fargle. What's the asymptote of $x^{1/3}(1-x)^{2/3}$?

Give me a few minutes here. I'm not in math mode yet. :P

LOL ... goof-off mode?

Driving for five hours to a baby shower mode.

Oh, ugh.

Fun baby shower, though. I mean, it's my brother and his wife. One of few circumstances where I'd care. >_>

11:22 PM
Oh, well, in that case ...

Still, I had to DM a D&D session last night, so I only got 3 hours of sleep before dusting off toward Atlanta.

Uh huh. "Likely story"

It really is, isn't it?

Hey everybody!

Hi Demonark

11:24 PM
Howdy @Daminark.

How's everything going?

Well, I was doing fine before Ted gave me this puzzle. :P

C'mon Ted why do you have to do that to the poor fellow?

Fargle likes to hate Ted.
How was Ted supposed to know he was in baby shower mode?

I don't! I just like to give Ted a hard time.
But that's only because Ted likes to give me a hard time. Clearly.
:P
@Ted Note that $$\lim_{x \to \infty}\frac{\sqrt[3]{x^3 - 2x^2 + x}}{x} = \lim_{x \to -\infty}\frac{\sqrt[3]{x^3 - 2x^2 + x}}{x} = 1.$$
I claim the asymptote is $y = x$.

11:35 PM
But your claim be wrong :P

oh no
I guess I don't know the definition of an asymptote, then.

If you try graphing using Calc I techniques, you'll find yourself contorted in contradiction. That's how I discovered this years ago ...
You want $\lim\limits_{x\to\infty} f(x)-A(x) = 0$ when $A$ is the asymptote function.

Ahhh.
And I assume that, uh, I can't just say that the asymptote is $x^{1/3}(1-x)^{2/3}$.

Um, no. I give you a hint: $A(x)=x+c$ for some constant $c$. :)

Hmm. Alright.
Well, I can prove that $-1 < c < 0$, I believe.
...hang on. My calc 1 gears are betraying their rust.
Let me see. I think--and correct me if I'm wrong--that it stands to reason that $\lim_{x \to \infty} x^{1/3}(1-x)^{2/3} - x = c$.

11:53 PM
Sure @Fargle.

I'm looking for a expository paper that I've read a while ago. It was regarding trigonometric functions - rationality/irrationality, linear independence, algebraic values. It had some theorems about cyclotomic fields/subfields.
It was quite a long paper, and contained a ridiculous amount of information; aimed at undergrads I think.
Searching I've found papers by P. Tangsupphathawat (Algebraic trigonometric values at rational multipliers of $\pi$ ), J. M. H. Olmsted (Rational values of trigonometric functions), D. H. Lehmer (A note on trigonometric algebraic numbers). It's none of these. First the name Lehmer sounded very familiar but it isn't that paper, and I couldn't find another similar paper by him. It was easily available on Google search as well.
It appeared on a journal, Amer. Math. Monthly or Gazette, but it could have been another one. Does this sound familiar to anyone?

@Fargle: Perhaps you want to think about how you might use Taylor ideas for $|x|$ large rather than small. :P

Perhaps this is a good time to mention how poorly I remember Taylor's theorem.

Well, you only need linear approximation (i.e., equation of a tangent line).
But I was serious that I only discovered the subtlety when graphing the function was inconsistent with having $y=x$ as the asymptote.

Yeah, I've noticed the same discrepancy.

11:58 PM
Ah, cool.
I wasn't meaning to disrupt your partying for the evening. :)

Oh, no partying for me. I only know family down here.

Oh ;(