/sci/ - Science & Math

seeking generalisations of this to nxn grids, is there any literature i can look at?

[math]{\frak Complex\ Analysis\ Theorem\ of\ the\ Week}[/math]

Everyone knows about the Bohr–Mollerup theorem, which gives a real-analytic characterisation of the [math]\Gamma[/math]-function. Far fewer people, however, know about Wielandt’s theorem, a similar result in function theory.

[math]{\rm {\bf Theorem\ } (Wielandt).}[/math] Let [math]f[/math] be analytic in the right-half plane such that

1) [math]f[/math] satisfies the functional equation [math]f(z + 1) = zf(z)[/math] for all [math]z[/math];
2) [math]f[/math] is bounded in the strip [math]\{z \mid 1 \le \mathop{\rm Re} z < 2\}.[/math]

Then, [math]F(z) = F(1)\Gamma(z)[/math].

For a proof, corollaries and further information, see
R. Remmert (1996). „Wielandt’s theorem about the [math]\Gamma[/math]-function“. American Mathematical Monthly. 103: 214–220.

Is there something like a fundamental group of a ring?

>>12796724
Whoops... [math]F[/math] is [math]f[/math] of course.

>>12796760
what's the context that made you ask this question?

Brainlet question but I always get confused by the simplest fucking shit.

I have an optimization problem of the form [eqn]P(x) = \min_{y} f(x,y) g(x,y).[/eqn] Now, each of the functions is itself an optimization problem: [eqn]f(x,y) = \min_{a} h(x,y,a)\\g(x,y) = \min_{b} i(x,y,b).[/eqn] Can I just write [eqn]P(x) = \min_{y,a,b} h(x,y,a) i(x,y,b)[/eqn] or do the two problems have to be done separately?

I realize that this is either trivially true or trivially false, but that's precisely why I'm confused.

>>12796781
I have a result telling me I can do homological algebra in the following situation: Thinking of a group [math]G[/math] as a one object category, take a functor [math]F\colon \mathbf{C} \to G[/math] from a small and connected category into the group, inducing a surjection [math]f \colon \pi_1( |\mathbf{C}|) \to G[/math] and let [math]K = \ker(f)[/math]. If [math]H_1(K; R) = 0[/math] for a unital commutative ring [math]R[/math], there is some homology stuff to be done for the categories of over [math]R\mathbf{C}, RG[/math]. I know this can be done more generally than with a category algebra [math]R\mathbf{C}[/math] as a source, so I was wondering if I could take some ring/algebra [math]A[/math] and it would have some [math]\pi_1(A)[/math] which I could try to relate somehow to that setting.

Any recommendations on computational proof verification? I've been googling a bit and there's not much information. I want to do a small pet project where I can verify a simple class of induction proof which I'm kind of stumped where I would even begin to do that in computation logic. It's so simple when writing it down with pen and paper.
Basically y divides x if true print out a general proof format. If false print counter example

>>12796826
See the Automath Archives.
https://www.win.tue.nl/automath/

wake up babe, new wildberger dropped
https://www.youtube.com/watch?v=grN-nDNQs74

>wildeberger after the destruction of the last axiom of choice in the universe

can anyone recommend good literature/textbooks on linear temporal logic (and Buchi automata too if pos)?

>>12796788
I think you can only do so if both functions are nonnegative. Otherwise when you pull one of the mins to the front it becomes a max sometimes.
As long as both functions are nonnegative, you should be able to push f into the min inside g, then push I into the min inside f, and now you have 3 mins in front which should commute? I can't imagine how they wouldn't, after all for each fixed a the min in y will be above the min in both a and y.

Finished rudin pma through derivatives. Wife selected munkres topology as the next book at random from alhfors, dummit/foote,munkres,spivak on manifolds
I guess I'll do the first half and then move to hatcher if I want the ap stuff

>>12796760
Read sga1

>>12796788
>this is either trivially true or trivially false
these problems are the absolute worst

>>12796347
what is so good about these books? Is it just finance bullshit or are they actually worthwhile

>>12797323
I wish I knew French.

Anyone here knows statistics really well? Can you do my test that will start in 20 minutes for me?

>>12797627
Sure thing bro. Check your DMs, I sent you the solution manual.

So, I'm on my first semester of math and I can't prove for shit. There's just this one subject that expect us to prove for now but I have never done this before. Is there a good crash course so I can do well on my first test?

>>12797213
that's true, one definitely needs nonnegativity. but other than that, do minima always commute like that?

>>12797727
Where the fuck are you at that requires you to actually prove shit in the first semester? Harvard or Russia?

>>12796673

I got accepted to a PhD program recently. I don't have a master's degree and I'm graduating from my undergrad after my 3rd year in college. I used to be 100% sure that is what I wanted to do with my life, but that it wouldn't happen. Now I'm not sure about it because doing a PhD comes with a lot of bullshit:

> Teaching responsibilities (IF I get an assistantship)
> meager stipend (17k < 21k)
> bullshit credit requirements
> spending the next 5 years chained to a desk
> just to do a thesis that nobody asked for that probably has no value to humanity

I'm considering joining the military just for the experience. I have spent the last semester getting in shape. It's really weird how my priorities in life are suddenly changing now that it is time to actually do it. I think I'm going full circle like the Unabomber.

>>12796349
Nice blog
Where did you get stuck anyways

>>12797744
That should be everywhere.

>>12797744
where the fuck are you at that doesn't require you to prove shit in the first semester? lol

>>12797779

it's sad how low the standards are for math undergrads.

> Bro it's only the FIRST YEAR of my math degree, we haven't learned proofs yet!!

>>12796673
Why do I have trouble with studying math? Specifically I look at the theory and I understand it, but once I understand it, I have no interest in doing the exercises which would solidify that understanding.

>>12797813
You are a pussy

>>12797816
I see no relevance

>>12797813
>I have no interest in doing the exercises which would solidify that understanding.
Anxiety. You feel like you won't be able to do the exercises, so you'd rather stay in your safe space (the theory that you managed to understand). The truth is, understanding the theory means absolutely nothing. If you can't do the exercises, you didn't learn anything. The essence of mathematics is in its exercises. It's really 30% theory and 70% exercises. What you need to do is work on leaving your comfort zone by engraving in your mind that you still didn't learn shit because that's what is actually happening.

I'm trash. I'm gonna fail another subject. I'm so fucking scum. I'll never finish my math major at this rate. I wanna cry.

>>12796826

The thing about proof assistant software is that it is totally alien compared to imperative programming and conventional mathematics. If something sounds easy in normal mathematics, it is very likely not to be easy in Martin-Lof type theory (or whatever specific theoretical system your proof assistant uses.)

When a theorem gets formalized on a computer, it usually requires devising a totally different proof that sheds a new light on the theorem. Formalizing a theorem on a proof assistant requires thinking so hard about the theorem that it crashes the operating system of your brain 5 times.

Go ahead and try it, see what I mean

>>12797580
Reading mathematical french is just a matter of willpower if you speak english.

>>12797836
>What you need to do is work on leaving your comfort zone by engraving in your mind that you still didn't learn shit because that's what is actually happening.
This is just the worst part, as there's a kind of a block in my mind which prevents me from doing so. The anxiety level shoots through the roof and I would rather fight an alligator than continue doing this.

>>12797902
Then you should seek medical help. There's no shame in doing therapy if your problem is this big and you can't overcome it alone.

>>12797902

NO, you must fight your natural instincts and submit to industrialized basedciety

>>12797953
Wait, are you telling me not to do that?

>>12797896
It seems surprisingly readable.

>>12797996
English came from Old French. They're virtually the same language.

>>12797744
Everywhere? If you're a math major and you're taking shit like calc 1 for engineers you aren't going to make it.

>>12798012
Not my native though. As someone who has no idea about schemes, can I use a noncommutative ring as one?

>>12798012
My old english teacher used to say English is 80% French vocabulary and 90% German grammar

>>12797727
The only way to get better is to practice a lot. Read your proofs out loud and try to reason that way. Write heuristic arguments and then turn them into formal proofs. Learn the proofs of theorems you do in class in a lot of detail, and pay attention to the turns of phrase and the intuition behind the proofs. Most of all you will get better with experience. I had no clue what I was doing in my first semester proof-based courses as well.

>>12797734
Sure, they must (I was weirded out because maxima and minima don't, but this is the same as the negativity issue)
Why? Well, min_a min_b f(a, b) is clearly smaller than min_a f(a, b) for each fixed b, since min_b f(a, b) is smaller than f(a, b) (we just took the min over a of both sides - easy to show this preserves order). but if min_a min_b f(a, b) is smaller than min_a f(a, b) for each fixed b, then it's smaller than min_b min_a f(a, b) since min_a min_b f(a, b) is a constant that doesn't depend on b any longer. You can then switch a and b to get the opposite inequality, which proves that mins commute (does the same argument imply infs commute?)

>>12797760
Here is the honest to god truth: If your perspective is "spending the next 5 years chained to a desk" and "just to do a thesis that nobody asked for that probably has no value to humanity," then you have no business doing a PhD program in mathematics. if there's a question in your mind whether or not this is the right choice, then you will be beaten to a pulp by a PhD program. also, that stipend is not a very good stipend (depending on the living costs of the area of course).
i highly recommend you don't jump into a phd program if you're not fully committed. you probably have more than a month to ponder on this. how much research experience do you have? have you taken graduate coursework? if you haven't got much of either of these then you probably have no clue what you're looking at.
there is absolutely nothing wrong with deciding to do something else. you can still enjoy math and learn it in your free time. you are also perfectly welcome to apply to masters programs later, get a masters, and if you enjoy that it will let you jump back in to get a phd.

>>12797790
it's because a bunch of idiots think they're going to make it in math (they won't) if they're taking fucking calculus in their first year. who the fuck didn't do AP calculus in high school or an equivalent. jesus.
>>12797813
sounds like you need discipline. exercises are fun, maybe you need to force yourself to start doing some and build some momentum. building habits of doing problems yourself / verifying unverified claims in a book is really really important if you're not being forced to by a course.

How do you all feel about curves that don't apply to people who did well. I got basically the highest exam grade in my class (adv lin alg), where the average was roughly a 55. Professor said people who got below an 80 can resubmit for up to an 80 (so this includes the lazy fucks who deserved to fail). It just bothers me that I don't get a curve, if even a slight one because I did too well (I got an 84 w/honors grading, 90 w/regular grading). Should I email him and ask for maybe a chance at a slight curve (maybe regain 50% of points loss for up to an 92?), if even not as substantial as the fucks getting a 60 point curve?

>>12798290
>Sharon Stone
I'm not a coomer, but damn does she get my engines rumbling

what’s the best book on measure theory

>>12798463
Folland

>>12798463
Sorry brainlet here, but why does one need a whole book on measure theory? My knowledge of the subject is probably limited to what's covered in Durrett or the like. What am I missing?

>>12798032
No, the ring needs to be comm. for the prime ideals to form a well behaved space.

>>12797744
lol

>>12797896
This. I highly recommend spending a few months studying French because after that point reading maths papers or books (with the occasional help of google translate) is a breeze.

You know, it's sad that French isn't the lingua franca of mathematics. It's the superior language for writing mathematical prose.
(I'm not a frog, by the way. Not an Anglophone either.)

>>12798816
>It's the superior language for writing mathematical prose.
No such thing. Math is a language in and of itself

>>12798824
Have you ever read a mathematics paper? A monograph? A textbook? They're not all formulæ...

>>12798290
>honors grading
>regular grading
wtf? something is either right or it's wrong.
also, who gives a shit if it won't affect your final grade. stop comparing yourself to others. if it will affect your final grade, then maybe mention that to a TA/professor.

>>12798816
French and English are nearly identical when it comes to math writing, not sure what you're talking about.

>>12798463
halmos is the bible of measure theory. folland real analysis would be fine too. or stein and shakarchi real analysis.
>>12798733
abstract measure theory (the theory of measures on general measure spaces) is pretty intense and deep. durrett covers only the measure theory required for a basic study of probability. halmos covers not only all the essentials of measure theory in great detail but also the theory of haar measures on locally compact groups, which is a very deep and important theory.

>>12798842
honors do an extra (hard) problem.

Has the set of all points constructible on R2 with compass and straightedge been proven to be exactly some other set like (Q/sqrt2)2 (which it definitely covers at a minimum) or is still unknown what set it exactly corresponds to?

>>12798816
Ich fand deutsche ist besser, mehr prazis und kurz

lol

>>12799151
Oh god
Oh fuck

https://www.youtube.com/watch?v=JHtvAZaPbK4

>>12799151
haha

>>12798978
Replace R2 with C, then showing that the compass and straightedge constructible points give the maximum iterated quadratic extension of Q is an easy excise.

>>12799045
>kurz

>>12798012
English is Germanic since saxons came from Germany. normans took over so they adopted a lot of French vocabulary.

>>12800003
Was?

>>12799970
Why is it easy to show that you don't get more than that?

[math]\frak On\ Factoring\ The\ Sum\ and\ Differences\ of\ Powers\ [/math]

I have a couple of questions related to the title of this post (1st line).
First, is there a general way to factor [math]a^n+b^n[/math]? I believe I've seen one for when [math]n[/math] is odd but none for when [math]n[/math] is even, why?
How do you prove: [eqn]\begin{split}
a^n-b^n &= \sum_{j=0}^{n-1}a^{n-j-1}b^{j}\\
&= (a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+a^{2}b^{n-3}+ab^{n-2}+b^{n-1})
\end{split}
[/eqn]
I got filtered by proof wiki
Finally, why do people separate [math]a^n-b^n[/math] into two different equations for odd and even [math]n[/math]s?

Re-posting here because it's taking a while for my question to be answered in the other thread (can't delete it either unfortunately).

>>12798744
O thanks friend.

>>12800658
>How do you prove...
How do you think? Multiply it out and you'll get a^n - b^n.

https://www.math.columbia.edu/~woit/wordpress/?p=12220

>>12800682
Nevermind, you're post wasn't useful but I think I got it lol.
Now can somebody answer my other questions?

>>12797876
Anon I feel for you, I've been there, and I know how demoralizing it can be, but you have to keep moving forward, and push yourself.
I started forcing myself to go to office hours and talk to my TAs and classmates if I didn't understand anything, and I would stay there until I understood it. And eventually I did understand it, and I got slightly better, and more confident. Suddenly, I could manage my course work, tests, projects, and everything else.
You have to put in the work Anon, as hard as it is to start. Once you get the ball rolling, you will look back on those long days of studying and work and realize how productive they were, and how they really weren't that bad.
Please do this, godspeed.

>>12800742
if n is odd, then
a^n+b^n=a^n-(-b)^n
if n is even then you need complex numbers

>>12800732
What do I need to do if I want to understand the IUTT papers?
Within the next seven or eight years.

>>12800755
Are you sure? Isn't [math]a^n+b^n, \rm where\ n\ is\ odd[/math] supposed to be equal this:
[math](a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-\cdots+a^{2}b^{n-3}-ab^{n-2}+b^{n-1})[/math]
This is obviously not what you posted but I'm sure I misunderstood you.
Also what do you mean by your complex factors claim? Please explain.

>>12800906
To clarify,
[math](a+b)(\underbrace{a^{n-1}-a^{n-2}b+a^{n-3}b^2-\cdots+a^{2}b^{n-3}-ab^{n-2}+b^{n-1}}_{\rm signs\ alternate\ between\ +\ and\ -})[/math]

Uh oh brehs... Feeling apotheotic again

>>12800987
kys mult

Bumping so some guy can finally answer my question.

>>12801020
If you can provide a good enough reason , I might, otherwise Ill assume youre retarded

>>12800658
>Finally, why do people separate an−bn into two different equations for odd and even ns?
What does this mean

>>12801091
You already asked it in /sqt/, which is where the question belongs, do your bumping there.

>>12800906
>this is obviously not what you posted
jesus man, take what I wrote
a^n-(-b)^n
now you expand it out using the very same equation you originally had
(a-(-b))(a^{n-1}+a^{n-2}(-b)+...)
and notice that (-b)^k is just b^k if k is even, and it’s -b^k if odd, which gives the alternating expression. For even n I think it would be best to get help in person

>>12801144
[math]{\rm let}\ {\mathbb F}\ \rm denote\ one\ of\ the\ standard\ number\ systems,\ that\ is\ \mathbb{Z, Q, R}\ and\ \mathbb C.[/math]
People seem to separate the general case ([math]n \in {\mathbb F}\ {\rm such\ that}\ n \geq 2 [/math]) into two special cases for odd and even [math]n[/math]s.
https://proofwiki.org/wiki/Difference_of_Two_Powers
>>12801243
Nobody answered it there and I assumed there would be more genuine mathematicians here. I wasn't trying to spam though, I just wanted answers (and like I said I tried to delete my last post in that thread).

>>12801302
taking a factor of (a+b) out you mean? look more carefully at the powers
multiply it out like anon has said, your question has been answered

>>12801290
Oh, my bad I get it.
The sum is just the general case except b is negative ([math]--[/math] is the same as [math]+[/math]).
I should have done it that way in the first place but I wasn't thinking I guess.
And I think I see what you mean with the complex numbers bit.
Sorry for being retarded anon, i'm learning, hehe.
I'll try to write it out completely in my notes, thanks anon.

>>12800658
>Finally, why do people separate an−bnan−bn into two different equations for odd and even nns?

Consider a polynomial [math]f(x)=x^n -y^n[/math], for some constant y. Since y is a root of this polynomial, you can divide it by (x-y) (Factor Theorem). For polynomial [math]f(x)=x^n+y^n[/math], we can say that -y is a root of f(x) only for odd n, and thus divide by (x-(-y))=(x+y).

>>12801314
The last question was just me wondering why there was a convention to separate the general case into two special cases. I wanted to know if it was easier or something. The important questions I had where answered already though thanks.

>>12801324
Thanks, I'm writing out the gist of this in my notes.

does anyone know where can I find online exercises for things like ring theory, or analysis?

>>12801302
I find this question very interesting. A cursory skim shows the proofs are similar in structure, so it may be that theres a lack in notation to refer to some antisymmetric thing with odds and evens

>>12801395
Like those randomly generated questions Wolfram has but for analysis instead?
Doesn't sound very practical, why not just do the exercises they have in textbooks like Tao's Analysis I, and II?

>>12801423
it is just taking a factor out which can not be guaranteed for odd powers
letting y=1 will make this very obvious

>>12801395
Not online, but Aluffi's Chapter 0 has a lot of good exercises for basic Ring Theory. I'm not trans so I don't know about Analysis

>>12798832
If you did read any of those you'd know there are several of them written in french. This is not the middle ages where everything was written in latin.

>>12796673
why is he dirty?

>>12801758
Because taters grow in the dirt...?
Do cityfolk seriously not know this?

>>12796673
Why is Von Neumann depicted as having a potato head?
>>12801771
Sneed

>>12801743
Yeah? I have read several of them in French, I said it's a shame it's not the lingua franca. Please apply some reading comprehension...

>>12796673
Hello my smart smart friends.
I need some help in this complex analysis problem, can someone please offer a hand?

Let [math]\alpha \in \mathbb{R}[/math] and [math]w \in \mathbb{C}[/math]. For [math]z \in \mathbb{C} \setminus H_{\alpha}[/math] we define the [math]w[/math] power of [math]z[/math] as:
[eqn]P_{\alpha}^w (z) = e^{w \log_{\alpha}(z)}[/math]

Now, consider [math]f(z) = P_0^{1/3}[/math]
(1) Compute its definition domain [math]A[/math] and the image set [math]B = f(A)[/math]
(2) Let [math]g(z) = z^3[/math] Find out if for all [math]z \in A[/math] it holds that [math]g(f(z)) = z[/math]
(3) Find out if for all [math]z \in B[/math] it holds that [math]f(g(z)) = z[/math] What about for all [math]z \in \mathbb{C}[/math]

For (1) Im guessing that the domain is [math]\mathbb{C} \setminus H_0[/math] but Im not sure about the image, i suspect its [math]\mathbb{R} \times (-\pi /3 , \pi/3)[/math]

the rest has got me completely stumped

>>12802056
motherfucker my retardation knows no limits ffs my mind feels broken

Let [math]\alpha \in \mathbb{R}[/math] and [math]w \in \mathbb{C}[/math]. For [math]z \in \mathbb{C} \setminus H_{\alpha}[/math] we define the [math]w[/math] power of [math]z[/math] as:
[eqn]P_{\alpha}^w (z) = e^{w \log_{\alpha}(z)}[/eqn]

Now, consider [math]f(z) = P_0^{1/3}[/math]
(1) Compute its definition domain [math]A[/math] and the image set [math]B = f(A)[/math]
(2) Let [math]g(z) = z^3[/math] Find out if for all [math]z \in A[/math] it holds that [math]g(f(z)) = z[/math]
(3) Find out if for all [math]z \in B[/math] it holds that [math]f(g(z)) = z[/math] What about for all [math]z \in \mathbb{C}[/math]

For (1) Im guessing that the domain is [math]\mathbb{C} \setminus H_0[/math] but Im not sure about the image, i suspect its [math]\mathbb{R} \times (-\pi /3 , \pi/3)[/math]

the rest has got me completely stumped

>>12798290
Are there any good systems for analysing things in knot theory that are continuously deformable to a circular loop like this? The definition of knots is that they aren't continuously deformable to circles so I don't even know what this type of thing is called.

>>12800789
The first step is reading the rebuttal by SS. The second step is understanding that there’s nothing of value in those papers.

Any logicbros here? I was perusing the CStheory stackexchange and found a post about realizability (can’t post the link because 4chan thinks it’s spam), and I’m thinking of going through the first two books in the first answer:
>Paul Taylor's book Practical Foundations of Mathematics
>Wesley Phoa's notes An Introduction to Fibrations, Topos Theory, the Effective Topos, and Modest Sets
Link for the second thing here (in PS format):
http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-208/ECS-LFCS-92-208.ps

Has anyone read Paul Taylor’s book? How is it? I've read logic up to Enderton and some Shoenfield, have about a year's worth of algebra (up to Galois theory), and have only done category theory in algebraic topology, if that's relevant.

I am trying to solve this exercise in Algebra:
(A) Find a field [math]L[/math] such that [math] \mathbb{Q} \subset L \subset \mathbb{Q}(\sqrt{3+\sqrt{2}}) [/math]
(B) What is the min polynomial of [math](\sqrt{3+\sqrt{2}})[/math] over [math]L [/math].
(C) Are there any other fields that satisfy:
[math] \mathbb{Q} \subset K \subset \mathbb{Q}(\sqrt{3+\sqrt{2}}) [/math]

For (A) I believe [math] \mathbb{Q}(\sqrt{2}) [/math] suffices. But I do not get how to take a min polynomial over such a field. Is it simply trying to find a polynomial with coefficients in [math]\mathbb{Q}(\sqrt{2})[/math].

Notation note: [math]\mathbb{Q}(\sqrt{2})[/math] means the extension of [math] \mathbb{Q} [/math]

http://www.strawpoll.me/42754768

>>12802521
In the literature this is whats known as a knot knot

>>12802888
Yes. I think you should be able to check x^2-(3+sqrt(2)) is the min poly over Q(sqrt(2)) by showing sqrt(3+sqrt(2)) is not of the form a+b*sqrt(2) for any a,b in Q

>>12803009
Ahh yes, this makes sense. Thanks fren

>>12802087
Someone pls :(

SURVEY

age?
single/married?
income?
job description?
highest level of education?

asking for a friend

>>12803520
22.
Single.
0.
Math undergrad.

>>12803520
26, married, 50k usd after taxes, but I live in the third world, teach calculus and mechanics, magna cooom laude from fancy all male liberal arts school

>>12803557
>all male liberal arts school
big gay learnatorium then

>>12803520
22
single
32k stipend
math phd student
bachelors degree, phd in progress

>>12797760
> I think I'm going full circle like the Unabomber.
same bro wtf

>>12803520
19, barely shy of 20
Single
Zero
Math student
Lurking /mg/

>>12798463
Cohn
Meme answer is the five volumes of Fremlin

>>12803708
Same bro

Is countable additivity in measure the same as lienartity?

https://en.wikipedia.org/wiki/Non-measurable_set

Frowin dis out deya

>>12803520
23
Single
32k pa when I start my PhD this fall, but about 6k pa in previous years
Last year of undergrad soon to start PhD

>>12803881
>BTP as a gold ball
i always imagined them/it as being hollow

How can i craft a bijection the (0,1) interval of real numberinto the (0,1)x(0,1) square on R2?

>>12803867
Lienartity?
How can a measure be "linear"? You can't multiply sets in a sigma algebra by a scalar. Anyway, linearity usually refers to finite additivity like mu(A disjoint union B) = mu(A) + mu(B) which is not the same as countable additivity.

>>12804281
Unzip. Put the even digits in one number and the odd digits in the other.

>>12804299
.090909... would then give the pair (0,1)

>>12804571
Ugh. Fine. Whatever. You can biject sequences of 0's and 1's with (0, 1), and you can biject sequences of 0's and 1's with pairs of such sequences by unzipping. Who gives a shit about the dumb 0.9999... crap anyway.

>>12803520
24
single
230k usd pretax
investment banking
math undergrad

>>12805151
If you take this approach you’re not answering the original question but you could probably just find a bijection from the interval (0,1) (not the ordered pair) to the reals, then unzip the reals into R^2, then map those back to (0,1)x(0,1)
another method would be to find a bijection from [0,1] to (0,1) but those are incredibly painful to deal with

For anon who self study math, how do you pick a problem? How do you know when to move onto the next topic?

>>12804295
But doesnt finite additivity imply countable additivity?

>>12804281
Given a number in [math](0,1][/math], break its decimal representation into complexes, i.e. sequences of digits [math]d_1d_2d_3\dots d_n[/math] where all but the last digit [math]d_n[/math] is zero. Interleaving complexes gives you a bijection.

Anyone else visualize projective geometry on the walls? Trippy shit man

>>12805157
>230k
How does it feel to have financial security
Genuinely asking