ok, I think there was some error in the derivation for the density we arrived at yesterday, namely $\frac1{2\pi}\frac{1}{x^2+y^2}$. I think it should be $\frac1{2\pi}\frac{1}{1-x^2-y^2}$. The second fraction times $dxdy$ is apparently just the area element of the unit sphere, hence my investigation into $\sin\theta d\varphi\land d\theta$ and how it becomes $\frac{1}{1-x^2-y^2}dx\wedge dy$.
@psie but the more I think about this problem, I really believe they meant the ball, i.e. they write "within the unit sphere". Otherwise they would have written "on the unit sphere"
@SineoftheTime plane equation is $-x+y+2$ the coordinates of $r$ are $(2,0,0)$ so $-2+0+2 = 0$ the equation is true, so the normal vector that i found is correct?
@Jakobian Many wastewater treatment plants in the world already remove the solid waste, treat it, and use it as fertilizer.
I'm struggling to find examples at the moment, but I do remember this being part of the tour of a waste water treatment facility that my 6th grade class went on in the early 90s.
Reuse of human excreta is the safe, beneficial use of treated human excreta after applying suitable treatment steps and risk management approaches that are customized for the intended reuse application. Beneficial uses of the treated excreta may focus on using the plant-available nutrients (mainly nitrogen, phosphorus and potassium) that are contained in the treated excreta. They may also make use of the organic matter and energy contained in the excreta. To a lesser extent, reuse of the excreta's water content might also take place, although this is better known as water reclamation from municipal...
Let T be a linear operator on a vector space V over a field. Prove that every one dimensional T-invariant subspace is an eigenspace of T.
I have shown that if $W$ is a one dimensional T-invariant subspace with basis $B=\{v_1\}$ then, $B\subseteq E_{c}$ where, $c$ is a scalar satisfying, $T(v_1)=cv_1.$
@Jakobian ok, I won't bother you more with this problem, but thanks for the help :) I think it turns out to be quite easy actually to find the marginal of $(X,Y)$ now that I've thought some more about it. We want to find the function $f(x,y)$ such that $$\iint f(x,y)\,dxdy=\iint \sin\theta \,d\theta d\varphi =1,$$ so we see that $f(x,y)$ is simply the Jacobian of that transformation (I guess in this case there's only one such $f$, so we are in no trouble)
@Jakobian Oh, what a mess! This question was given in one of our university exams. I got a printed copy of it. But I think it's pretty much the same what you say. I think the student didn't write the correction in the paper
This is the screenshot of that question for reference. I was talking about the part $(b)$
Sorry @Thorgott, I've been away a few days so only just seeing your comments now. At any rate, why aren't sequences defined like this! Makes much more sense
These are the last two questions I still can't manage to solve from the current section
I know we've discussed them, but I've as yet not been able to come up with a solution
For 6, I have almost no clue how to proceed in getting a bound
For 7, the hint is clear, but I want then to be able to demonstrate some fixed $M$ such that $M \leq \sum_{k=0}^{q-1} Y_n^kY^{q-1-1k}$ so that, on taking reciprocals, I can bound $y_n - y$. But it's not clear how to come up with that fixed $M$ for me
@SoumikMukherjee and @robjohn Thanks! I was able to solve it. But can you please explain what was your motivation behind this? Was this: math.stackexchange.com/q/817230/1093844 theorem a standard one?
obliv: i dunno, i don't think that is a very useful question (note that most algebraists do not spend much if any time contemplating the 'hierarchy of algebraic structures,' there is more than enough in the axioms for the non exotic stuff to keep anybody busy for life)
@EE18 the function $\text{Id}_{\mathbb{K}}(x) = x$ and constant functions are continuous. Moreover, sums and products of continuous functions are continuous. So any polynomial is continuous. And quotients of continuous functions are continuous on their domain. So any rational function is continuous on its domain
Another apology owed to you Jakobian: I've not yet seen continuity. I've only seen the definition of limit of sequence. I can include a few of the useful facts I've seen but I'm sure you can imagine a treatment which treats sequences before limits of arbitrary functions
I've seen that convergent sequences form a subalgebra of $\Bbb K^\Bbb N$ so I can easily extend that as required to show that this result holds for polynomials
concept of continuity is more general but for your purposes you should accept this as definition of continuous function
Prove that when $f_1, f_2:A\to\mathbb{K}$ are continuous then so are $f_1+f_2, f_1\cdot f_2$ and the function $f_1/f_2$ defined on $\{x\in A : f_2(x)\neq 0\}$
this should be obvious from already proven properties of sequences
I'll remark that this is Heine's definition of continuity
Whereas Cauchy's definition is about $\varepsilon$s and $\delta$s. Both are equivalent under axiom of choice
Let $f:\mathbb R^n\to N$ be a smooth function and $M=f^{-1}(p)$ be the preimage of a regular value (more generally we could have the preimage of a submanifold transverse to $f$) and we can assume $M$ is compact, oriented, and whichever mild properties that can make the question easier. The usual ...
I know you haven't defined continuous functions in your book yet, but this is more about what's convenient. You can treat it as us calling those type of functions that preserve limits something. Call it "property H" if you don't like calling them continuous. Doesn't matter @EE18
Consider a partition of $\Bbb R^3$ by topological surfaces $M=\Bbb R^2-\lbrace \mathrm{1~point} \rbrace$. Is it possible to make these surfaces $M$ complete, hence giving a partition of $\Bbb R^3$ with no singularities?
Understood Jakobian. I am going to first prove (though it should be easy) that the claim in the Exercise is true for polynomials. But I think it's the $f_1/f_2$ case which I still can't see how to argue. I feel like I'll need to get down into the epsilons and deltas but I can't see how
Anyway, will report back. Thanks for the start :)
(If you get the chance for a hint on the other question I would also greatly appreciate :)
OK one fact which I need (but don't want to go through proving because it seems to be a fact from algebra)
Suppose that we have defined $r(x)$ for $r \in K(X)$ as $r(x):= p(x)/q(x)$ for any $\langle p,q \rangle\in r$ with $q(x) \neq 0$ (and let's say we've shown this is well-defined, i.e. independent of equivalence class representative obeying the property).
OK, so suppose we're at an $x$ where such a $\langle p,q\rangle$ exists. My question is: it follows that in particular for the particular unique $\langle p',q'\rangle \in r$ in lowest terms (defined as $q'$ monic and of least degree) we have $q'(x) \neq 0$ right?
Yes $K(X)$ is defined as the field of fractions for $K[X]$ right?
I agree, that's how I'm defining said evaluation above. I bring it in because that's how my book has defined things. K(X) is the abstract field of fractions
OK, I will take what I wrote above as correct because I'm fairly sure it is and now get back to thinking of the function itself. But I guess my question is exactly about that relation
Was pretty simple, followed just from the correspopnding conclusion for polynomials (not sure why my book asked me to get the polynomials as a corollary of this given that I had to use it as part of the proof but anyway)
If you have any hints for 7 I would greatly appreciate :) im surprsed i couldnt find with a google search...
Two more questions in the meanwhile if possible :)
1) For this one, $A,B$ are easy. But what of $C$? I feel liek I should be able to find some clever factorization such that it becomes $(z-a)(\overline{z}-a) = (z-a)\overline{(z-a)}$ for some real $a$, in which case it's another one of these "distance from a" situations, but I can't see one
2) I am trying to find the set of solutions to $z^3 = 1$ without using De Moivre's formula or the polar form of complex numbers (because these have as yet not been introduced in my book). Is there any clever way to do this without brute forcing the system of solutions induced from $(x+yi)^3 = 1$? I managed to find a clever-ish way for $z^4 = 1$ in reducing it to the case for square roots which I'd seen
Can I shamelessly ask for a hint on that Q7 we discussed way above too?
lmao you read my mind
Yes $r$ natural is easy. I can solve the problem immediately once I know the situation with $r = 1/q$ because products of convergent sequences are convergent
I just can't solve that one case because i can't figure out a fixed lower bound on the sum I noted above
Sure, that was proved in a Proposition I saw earlier. That's the case $1/x_n$ for $x_n$ convergent to nonzero limit (and convergent to a positive number is enough to use this Proposition)
@Jakobian This is what I came up with (feel free to ignore all of it, but especially everything before the last two paragraphs). is this what you had in mind?