/mg/ - Math General

shit, [math]\forall \alpha \geq 0[/math] **

they call them "reals" but they aren't
really makes you think

why the fuck is the Collatz conjecture so fucking difficult?

"Mathematics may not be ready for such problems." -Erdős

I wonder if we'll be able to ever have a thread where no one will fall for that retard's "what axiom do I need to disprove X" bait, and similar.
>the real numbers aren't real
^this right here, this is what cancer memes look like.

Part of what makes it hard is that it doesn't really directly belong to any field, so good ideas for even starting on it are scarce right now.
A concrete problem is that relating the factors of 3n+1 to n (which determines how much dividing by 2 will shrink the number) is extremely hard because prime factorizations are little fuckers when you start adding numbers to them.

It's like the z transform

EEs are brainlets

It is just one of those problems that shows how little we actually know about numbers. We have to think of this in relation to Fermat's Last Theorem. At its surface Fermat's Last Theorem looks a theorem about numbers but we found out that actually it is a theorem about elliptic curves. There is no way to crack Fermat without first cracking elliptic curves. Similarly there must be some unknown mathematical object that actually determines the behavior of Collatz' sequence and I think that is what Paul meant when he said the quote you posted.

fug. sucks to be a brainlet, this stuff really bakes my almonds.

I'm not really sure what summing using a non-Integer starting point means. I guess that's what you're asking. Maybe it means do a, a+1, a+2, ..... But then why write it like that? Why not write it as Sum from n=0 to Infinity and replace n with n+a? Which would just get rid of the -a's.

I think there's something wrong with the way that problem is written.

You cannot use induction on the real numbers because [math] \mathbb{R} [/math] is not well-ordered (and you cannot define a well-order on it that is compatible with the natural ordering of real numbers) and no real number has a "successor" (and you cannot define a successor function that is compatible with all the other properties of the real numbers).
Seriously, fuck off with these stupid questions.
Because it's a halting problem.

>sqt
they couldn't give a serious answer if you paid them you anime posting degenerate loser.

>they couldn't give a serious answer if you paid them
I very much doubt that. Either way, this sort of stupid shit doesn't belong here.

The exponential isn't an eigenfunction of the fractional derivative, these are though
en.wikipedia.org/wiki/Mittag-Leffler_function

can a non-brainlet please peep this and tell me if its legit or bs

It's just the series expansion of sin(x)
en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions

please be joking

i was talking about the paper in the link i was replying to:

what's the best number theory book for complete understanding?

It doesn't look right to me but I could barely get any sleep last night and I'm feeling pretty groggy so I didn't read it in detail.

No one has complete understanding of number theory, and consequently, no book that imparts complete understanding of number theory exists.

Also,

> real valued index of summation
> real valued degree of the derivative operator

If this problem made sense, you could solve it for a in terms of the other variables. Is that even possible? I don't know

In fairness, it wasn't that good of a joke to begin with. As for the paper I skimmed it and it looks to be the same as most "proofs" of the riemann hypothesis, I haven't looked at it closely but I think there are some issues with in some of the arguments about applying the fourier transforms tom some J operator don't seem rigorous to me

w-what

What's the matter brainlet? You can't do a simple factorisation?

your notation is shit senpai

summing from n = a, then everywhere you use (n - a)? you can simplify it and write it in terms of standard summation notation.

you dont even need induction to prove that.

[eqn]\sum_{n=\alpha}^{\infty}\frac{x^{n-\alpha}}{\Gamma\left(n-\alpha+1\right)} = \sum_{n=0}^{\infty}\frac{x^{n}}{\Gamma\left(n+1\right)} [/eqn]

[eqn]\frac{e^x}{(1 + e^{x})^2} = \frac{e^x}{1 + e^{x}}\frac{1}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\frac{1 + e^x - e^x}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\frac{1 + e^x - e^x}{1 + e^{x}} = \frac{e^x}{1 + e^{x}}\left( \frac{1 + e^x}{1 + e^{x}} - \frac{e^x}{1 + e^{x}}\right) = \frac{e^x}{1 + e^{x}}\left( 1- \frac{e^x}{1 + e^{x}}\right) [/eqn]

slut

The "type" of naturals doesn't exist either though.

based user, thanks.

What are the best available foundations which don't believe in fairy tales such as the "real" numbers and other similar fabrications?

>I wanted to prove that [math]\forall \alpha ∈ \emptyset[/math]
That's pretty simple actually.

brainlet

Excuse me?

You're excused.

Kill yourself and rid us of your shit meme.
Fucking crank.

I don't expect you to understand.

why did we pick such a shit topology to be the "standard" topology on R

Any topology on "[math]\mathbb{R}[/math]" will be shit.

Can most rational numbers even be shown to exist? I'm starting to doubt the existence of most of them.

Let's suppose I believe in the existence of the naturals but not the integers. How should I go about formalizing this?

You'd need to refute the existence of Grothendieck groups.

Hm... This indeed seems like an obstacle. Can I at least "stop" at the integers, so to speak?

I'm doing real analysis + group theory right now but I seriously don't know what it's useful for

>real analysis
Garbage which should be completely ignored.

Can the mods fucking ban this real numbers aren't real fag already?
Please?

Reddit might be a better website for you.

group theory is useful for improving your cubing speed

would a nonstandard topology be more appropriate for analysis on the rationals?

daily reminder that post 18th century math is brainlet cancer bullshit
prove me wrong
protip:
you can't

>the rationals
It is not clear that there are any sets which are not finite.

Reposting from /sqt/ since there seem to be lots of group theory anons here

>No one has complete understanding of number theory
Speak for yourself please.

Math General cute boy orgy when

I have a believing heart and my believing heart believes you are a faggot.

You need help.

Think about what conjugate means

It's currently a conjecture.

>/mg/
>where continuum doesn't exist but gender is a spectrum

Gender is actually a discrete spectrum, not a continuous spectrum. Learn the difference brainlet.

For doing other mathematics. Take more statistics and computer science if you want something useful.

>statistics and computer science
Seems like the wrong thread for your kind. Please use something else for this type of garbage.

lebesgue integration for calc undergrad brainlet?
can I understand it or even just compute lebesgue integrals without measure theory prereqs?
I'd like to be able to compute the riemann integral of f(x), and the lebesgue integral of f(x) and be like "huh it's the same" or different, if it's different. I can't find anything about the topic that addresses brainlets, it's all grad level stuff.

>I believe in the existence of the naturals
what do you mean by this?

math.stackexchange.com/questions/4202/induction-on-real-numbers

If you don't know what an integral is beyond "take the antiderivative", it's not worth your time.

Veeky Forums, be honest, are you a classical math chad or a modern math cuck?

Do not waste your time with lebesque if you do not even know riemann. Learn the riemann integral and all its nice properties and tricks. It is actually more than enough for any elementary application. Lebesque integration is only necessary when you want to generalize your theory and be able to do integration on weird sets which you won't need for literally all of undergrad.

...

source?

>has never seen an irrational

Euler never saw his number?

read garden of integrals

I made it some years ago

Every set can be well-ordered, so there is well-ordering of [math]\mathbf{R}[/math] and therefore we can use induction on reals.

Choicefags aren't welcome here

>I'd like to be able to compute the riemann integral of f(x), and the lebesgue integral of f(x) and be like "huh it's the same"
the thought process goes as follows: you learn what an antiderivative is. then you approach the "area under curve" issue which is a completely unrelated problem. you define the riemann integral as an abstract tool for measuring the area under the curve. now you have the definition which you think makes sense, but you have no idea how would you actually compute the value. turns out that you can do it using the antiderivative. then you find out that the riemann integral has some issues which you think it shouldn't have, so you come up with a better tool for measuring areas, the lebesgue integral. now you have your better tool, but once again, you have no idea how to compute it. turns out, that the lebesgue integral equals the riemann integral provided both exists. so the lebesgue integral is also computed using the antiderivative.

never in your life you're going to compute a riemann or lebesgue integral using the definition (that might be an overstatement)

This is a really wonderful book if you're interested in this topic. Full formal treatment of the subject, exercises, well written, good notation, an all around pleasure.
I bought it last summer, went into it a bit and am just returning to it soon, after some work is finished this month. I can't wait!

I see, thanks a lot for the clarification.

Actually in calc 2 my professor made us compute the riemann integral of f(x) = x^3 from 0 to 1 and then asked us to compare with the antiderivative approach.

It was tedious but it was kinda fun.

lots of good fap material in this. recomended

I have noticed the funniest, most original posts happen from people who add extra spaces between their lines. I wonder if there is some kind of correlation.

Russel was BTFO so hard by Godel, he spent a fuck ton of time writing his meme book and then BOOM *super theorem*.

Also, where did he get the idea that Godel was jew? Fucking anglos.

this:

truly marvelous

>I don't know what group theory is useful for
>I don't know what the study of symmetry is useful for
You sir are a brainlet and a philistine.

>math.stackexchange.com/questions/4202/induction-on-real-numbers
Point to where a well-order on [math] \mathbb{R} [/math] is constructed or an analogue to the successor function is defined on it. All that guy does is construct a model for transfinite induction on intervals of real numbers that are bounded from bellow. [math] (0, \infty) \neq \mathbb{R} [/math].

kek good luck doing anything of worth in stats without having knowledge of real anal and measure meme.

Russell is a known charlatan among philosophers. He lied like crazy because he was a politically motivated loon. A proto-SJW so to say. See for example his book on the history of philosophy. It's full of crass inaccuracies that he inserted there for pretty much social justice reasons. "muh Aristotle justifies oppression so I'll paint him as an idiot by misrepresenting what he wrote and read his works in the most uncharitable way I can manage".

Read what I wrote again. Something important might dawn on you.

Is a function still considered well-defined if one output can have multiple inputs? Example, sin(x).

By definition, all functions are well defined. Well defined only means that when you write down an expression, it has one and only one value or interpretation. If you write in(1) it is equal to one and only one real number, there is no ambiguity and therefore it is well defined.

Guys, how to show that [math]F(A ⊔B) = F(A)*F(B)[/math], where [math]F(X)[/math] denotes free group over [math]X[/math], [math]⊔[/math] is a disjoint union, [math]*[/math] is a free product. I.e. I need to show that free groups preserve coproduct.
I can only show that they are isomorphic by messing around with universal property, but I have no idea how to show they are equal. Any hints?