Does the IVP $$e^x \dfrac{dy}{dx} = e^{y^2} \sin x - \dfrac{1}{y^2+1}, \: y(0) = 0$$ have an unique solution in a nbhd of origin?
If $f(x,y) = \dfrac{dy}{dx}$, do I have to determine analyicity of both $\partial f/ \partial x$ and $\partial f/ \partial y$ in a nbhd of origin or just the second?
I thought both partial derivatives (along with $f$) had to analytic, but it seems I'm wrong
I'm studying for my complex analysis final. Any didacts here have any common mistakes to watch out for? Cases where students often dont realize the hypotheses for a thm aren't fulfilled?
I wanted to generate a random subset of the set {1,...,n} with probability of each subset proportional to (n choose k). I figured that in order for this to work, $P(A)$ must be (n choose k) over (2n choose n)
Trying to figure out an algorithm out of this expression is driving me nutts!!
Quite often in probability, if you look at the probability each "object" must be yielded with, you can come up with an algorithm
For example, if you want a subset of {1,..n} proportional to the number if its subsets, P(A) must be (2^k / 2^n) (the size of the set is k) after normalization
And that clearly signals that what you do is generate a random string of length n over the alphabet {0,1,2} and then if you yield the string (e1,...,en) you take as a subset {i | ei in {0,1}}
(n choose k) over (2n choose n). What if we randomly sample a subset of {1,...,2n} (assume we can do that), and then take the first k elements that are in the range {1,...n}, if any
I wrote down that's impossible for a complex power series to converge in $D(a,r)$ and its closure except at exactly one point on the boundary $|z-a|=r$. , why is it impossible?
I meant that it converges absolutely in $D(a,r)$ and converges on $\mathbb C\setminus D(a,r)$ perhaps conditionally, and it diverges at exactly one point $z$ with $|z-a| = r$
I wrote down that that's impossible
but converges on, say, the rest of $|z-a| = r$
so that it's radius of convergence is $r$ because r is the supremum of $\{r \ge 0: \sum a_n z^n \text{ converges for } |z| < r\}$
anyway, I am going to get back to work. The sooner I'm done with this the sooner I can start on the massive pile of self-studying I've unfortunately assigned myself.
$G$ is simply connected $\iff$ $\pi_1(G) = 0$ $\iff$ every circle in $G$ can be filled in continuously into a disk $\iff$ every closed path is h.e. to the trivial path
Hm, there's a terminological flubberduck that's happening on my end
So I call the dynamical system given by the linear ODE $\mathbf{x}' = A\mathbf{x}$ hyperbolic at the fixed point at the origin if $A$ has no eigenvalues of zero real part.
nothing yet, I just want to know if i can feed a power series into any theorem that eats a laurent series. It just has all the coefficients on the negative terms = 0 so I dont see why not
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You say The Who-i say Indie
You say Classics-I scream Alternative!!!
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I have to decide at what point i should take a break before the exam, if I study straight for another 3 and a half hours until the exam I might be mentally fatigued
I feel like starting at any morbid sad boi song and going through the rabbit hole of youtube recommendations is a good approach to knowing a lot of different musical genres which I am interested in
Like I started with breakcore and I am finding lots of generative ambient musics now
I need to find the area for the region bounded by y=2cosx and y = -sinX I have the graph but I need to find the x value when they intersect how can I solve for x given 2cosx = -sinx
So on part of the interval one curve is on top and on part of the interval the other curve is on top. They want the total geometric area between the curves on the whole interval.
