@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 $v$.
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
$-I$ 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 $n$
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…
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...
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
@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.
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.
@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.
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
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?
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
@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
@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"
@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 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.
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
@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)
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)
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.
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
and he asks us to take advantage of that fact
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?
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?
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...
