12:43 AM
@psie this is something from differential geometry I think. And you probably shouldn't bother yourself with it when learning probability

8 hours later…
9:07 AM
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 $\frac{1}{1-x^2-y^2}dx\wedge dy$.

9:49 AM
@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?

10:12 AM
@Pizza no it's not correct
@Pizza this is wrong
you can't conclude that the normal vector is $(-1,1,0)$
you only know that $(1,1,1)\cdot (a,b,c)=0$, i.e. $a+b+c=0$

I have to ensure that and that the plan passes through the point $r$, defined as $(2,0,0)$?

$r$ is not a point

10:28 AM
The plane for $r$ parallel to $s$ is a plane that contains the line $r$ and which is parallel to the line $s$?

yes
if it contains $r$, what is the relation between $(a,b,c)$ and $d_r$?

10:43 AM
@SineoftheTime the coefficient of the component $d_r$ of the normal vector $(a, b, c)$ of the plane equation will be zero?

is the triple (abc) of real values ​​called director numbers of the r plane?

r is a line not a plane

I'm getting confused, because I'm reading
this
Take a plane $\pi$ and fix three points on it
A, B and C etc
what does it mean I take a $\pi$ plan?

$\pi$ is the name of the plane
as $r$ and $s$ are names of lines

10:58 AM

aren't you searching the plane parallel to s passing through r?

yes, no i mean in the old exercise where there was no solution remember?
but now I have to find it

okok
theoretically
\begin{align*}
r: & \begin{cases} a_1x+b_1y+c_1z+d_1 = 0 \\ a_2x+b_2y+c_2z+d_2 = 0 \end{cases} \\
\end{align*}

11:15 AM
what are you computing?

I have to represent my line r in Cartesian form, right?

what exercise are you doing?

the same

@Pizza this?

but I saw that there is another way to do it
yes the point 1

11:16 AM
you already have the equation of r

yes

I want to write the equation of the plane sheaf that has r
im following this
It's almost the same as mine

it's not the fastest way

I assume what you were suggesting was faster?

11:20 AM
yep
denote $d_r$ and $d_s$ the direction vectors of $r$ and $s$
and $(a,b,c)$ the normal vector of the plane you're searching
what are the relations between $d_r$ and $(a,b,c)$ and $d_s$ and $(a,b,c)$ ?

What do you mean by relation

if the plane is parallel to s, what can you say about $d_s$ and $(a,b,c)$

a plane and a line are parallel if and only if they have the same normal direction

no

does it have to be two planes?
@SineoftheTime wait
For a line or vector $r$ lying on the plane, $d_r \cdot (a, b, c) = 0$.

11:39 AM
correct

For a line or vector $s$ lying on the plane (or a plane parallel to $s$) $d_s \cdot (a, b, c) = 0$

correct
now substitute since you know $d_r$ and $d_s$

$d_r = (1,0,0)$ and $d_s = (1,1,1)$ right?
now i can substitute

@psie in my derivation there was a moment where I changed cosines to sines. This is probably the place where I made an error

$d_r$ is wrong

11:45 AM
@SineoftheTime isnt r : x = 2+t , y = 0, z = 0?

21 hours ago, by Jakobian
one could substitute cosine I guess, it'd lead to something like $\iint_{(\rho \cos\varphi, \rho \sin\varphi)\in A} d\rho d\varphi$

why $y=0$?

@SineoftheTime y is constant and remains 0?

no
y is not constant

i would say then y=y

11:52 AM
?

I don't understand otherwise how I can form the vector
without y
in the equation of the line r, y is not there

$r: x=2, y=t, z=0$

can y take on any value?

$d_r = (2,t,0)$ and $d_s = (1,1,1)$ right?

12:01 PM
$d_r=(0,1,0)$

ah I understood how you did it, then I found the other vector correctly but I had made a wrong reasoning
when I write the parametric I consider the t as 1 while the numbers are 0?

yes
now substitute

$\mathbf{d}_r \cdot (a, b, c) = (0, 1, 0) \cdot (a, b, c) = b$
$\mathbf{d}_s \cdot (a, b, c) = (1, 1, 1) \cdot (a, b, c) = a + b + c$

and both have to be $0$

yes
$b=0$ and $a+b+c = 0$

12:05 PM
go on

but do these need to be in a system?

$a+b+c=0$
$a+c=0$
$a=-c$
$b=0$
so
$(1,0,-1)$

@Pizza for example
so your plane is $x-z+d=0$, now you only have to find $d$

$3-1+d=0 , d=-2$

12:11 PM
what did you substitute?

$A(3,0,1)$

$A$ does not belong to the plane
$r$ belongs to the plane

@SineoftheTime do I replace the values ​​of x and z with the values that are in the equation of the straight line r?

no
for example $(2,0,0)\in r$
so in particulal it has to belong to the plane

mm so $2-0+d=0$
$d=-2$

12:21 PM
yes

$x-z-2=0$
but to find $(2,0,0)$ didnt you consider the values that were in the equations of r?

just take any point $\in r$

$r: x=2, y=t, z=0$
for example $(2,1,0)$
would it always be okay?

then the y isn't even present
x−z+d=0 here i mean

12:28 PM
yes, the coefficient of y is zero

12:40 PM
Thank you so much for your time! @SineoftheTime

12:50 PM
np

1 hour later…
1:51 PM
If we were to figure out how to collect human poop in a hygienic way, then would it be possible to use it as fertilizer?
I think that would make humans more self-sufficient, and so efficiency overall would increase

@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...
2

2:12 PM
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)

I'd be done if I were able to show that $E_c\subseteq B$. But I'm having a hard time showing that.

@psie what you said makes no sense to me, but alright

@Jakobian Would you mind helping me a bit with the problem above?
I'm badly stuck with it :?)
@XanderHenderson I'd be glad if you consider lending me a helping hand, as well.
But I need to solve this :D

@ThomasFinley I do mind, but maybe I'll look at it sooner or later
@ThomasFinley alright I'm looking at it. Didn't you prove $v_1$ is an eigenvector with eigenvalue $c$ with this
also you want to prove $W = E_c$

2:29 PM
@Jakobian Yes, sure.

$B$ is just a set containing one vector

@Jakobian If $B$ is the basis for $W$ then yes.

that's what you wrote above, so it should be

@Jakobian yep

@ThomasFinley what are your definitions of eigenspace and T-invariant subspace?
$T(w)\in W$ for $w\in W$, and $E_c = \{x\in V : T(x) = cx\}$?

2:33 PM
@Jakobian Exactly

then what you are trying to prove is false
Let $T(x) = x$ for $x\in V$, $W$ be spanned by some non-zero vector $e$
then $W$ has all desired properties, but every element of $V$ is in the eigenspace corresponding to eigenvalue $1$
if $V$ is any space with $\dim V > 1$ then you get a counter-example

@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 $(b)$

@ThomasFinley I think the purpose was to show $W\subseteq E_c$ only
i.e. $W$ is contained in an eigenspace of $T$
b') Prove that every one dimensional $T$-invariant subspace is contained in an eigenspace of $T$

@Jakobian Yes, looking at the screenshot I feel the same and your reformulation $(b')$ for $(b)$ is perfect. Thanks!

1 hour later…
3:58 PM
Find an inner product <, > on $\Bbb R^2$ such that <(1,0),(0,1)>=3.
I have no idea how to solve this

4:13 PM
Anyone?

4:55 PM
How do logicians axiomatize different set theories/formal logic systems?
Do they rely on natural language in the end? I don't think everything is all derived from a single formal logic system so I imagine so

@ThomasFinley Find a suitable matrix using this

5:58 PM
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

@ThomasFinley $$\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}a&3\\3&c\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix}=3$$

6:36 PM
@leslietownes at what level of structure in the hierarchy of algebraic structures does there become a lot of utility?
I read on wiki that composition of functions can be seen as a composition monoid
but a lot of functions don't have the same domain so idk why they preface that

@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?

7:03 PM
@EE18 is $\mathbb{K}$ reals or complex numbers

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)

@Jakobian thank you for pointing that out. The convention in this book is that it is either

well you said people probably don't care about partial magmas/magmas so I was wondering at what point they care about the structure

@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

@EE18 what real analysis book is that again

7:06 PM
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
This is Amann Escher Obliv

@EE18 a function $f:A\to \mathbb{K}$ is continuous when for any sequence $x_n\in A$ with $x_n\to a\in A$ we have $f(x_n)\to f(a)$

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

$A\subseteq \mathbb{K}$

What I can't see is how to deal with the extension to field of fractions

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
doesn't really matter for you right now
what matters is how this all works conceptually

7:14 PM
0

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

topology question
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 :)

7:44 PM
@EE18 you should already have that $\lim_n (x_n+y_n) = \lim_n x_n + \lim_n y_n$ and $\lim_n x_ny_n = \lim_n x_n \cdot \lim_n y_n$ and $\lim_n x_n/y_n = \lim_n x_n/\lim_n y_n$ whenever $\lim_n y_n\neq 0$
you don't need any arguments with deltas and epsilons
@zetaspace by complete, do you mean complete as a subspace of $\mathbb{R}^3$
@EE18 maybe I'll look at it after you solve this one

8:03 PM
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).

@EE18 equivalence class?

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?

this is irrelevant to the question
you have a quotient of two polynomial functions $x\mapsto p(x)$ and $x\mapsto q(x)$

I promise it's partially relevant :) it's all part of the formalization of "evaluating" a rational function

why bring some irrelevant theory into this? Pointless
no, its pointless

8:09 PM
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

doesn't matter because a rational function as a function is not an element of K(X)
and I don't want to get into how the two relate with each other because thats irrelevant

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

OK I got it :)
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...

8:41 PM
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

$\|z-2\| = 1$

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

its a circle

OMG that's embarrassing, had to move stuff over to the other side...
Thanks Jakobian

@EE18 what are some obvious solutions

8:47 PM
1, of course
I can also use polar and kinda guess what the other two are
But hoping not to do that

and what do you do when you know one solution to a polynomial equation

AHAH! I knew there must be a clever way :) OK, I will run with this...thank you!

@EE18 suppose $r$ is natural. Can you solve it?

Can I shamelessly ask for a hint on that Q7 we discussed way above too?
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

what about $r = -1$, can you solve that
@EE18 I don't need all that information from you. I'm just asking questions for now

8:54 PM
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)

sure
oh sorry I got distracted
$$x^{1/q}-x_n^{1/q} = \frac{x-x_n}{(x^{1/q})^q+...+(x_n^{1/q})^q}$$
sorry got distracted again

@Jakobian Yup, agree with this. Just can't see how to bound the denominator from below so that its reciprocal can be bounded from above

so, sure, there is some $r > 0$ with $x_n, x\geq r$
@EE18 Find $N$ so that $|x-x_n| < x/2$ for $n > N$

9:10 PM
ok, i will think about both of these. not sure where it's going yet but will report back

9:48 PM
@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?