/mg/ -- Quillen Equivalent to Last Thread Edition

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Other urls found in this thread:

arxiv.org/pdf/math/0111082.pdf
en.wikipedia.org/wiki/Axiom_schema_of_replacement
en.wikipedia.org/wiki/Law_of_total_variance
r.amherst.edu/apps/nhorton/Adam-Eve/
en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
twitter.com/SFWRedditGifs

Anyone here good in propositional calculus?
I know this is a valid step
[math]x \Rightarrow x \lor y[/math] (weakening the concequent)
is this also valid?
[math]x \Rightarrow x \lor (x \land y)[/math]

add a "union y" at the end

sorry, what?

Yes it is. Implication is incorrect if and only if T==>F. So when x is true you have T ==> T v (whatever), so its true anyway.

Yes. You said you know it (although you obviously don't).
[math] x \implies x \lor (\text{anything}) [/math] is a tautology.

>propositional calculus
not math

How much does it usually take /mg/ to properly work through a textbook? I'm thinking something that would take a 1 (maybe 2) year course load.

x or (x and y) or y

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arxiv.org/pdf/math/0111082.pdf

Why though?

It takes forever and usually I don't want to know every last thing a textbook has to offer. This doesn't hold for the really short, specialized books, but for larger books I'm usually at this point looking to pick up specific knowledge and then learn something else.

What is a curve ? How do you measure the curvature of a curve ? Why do simple plane curves separate the plane in two regions ? How would you prove this for a convex polygon ?
Compare the length of a simple closed curve and its area.
Why can't R^n be homeomorphic to R^m if m n ? Can GL(n,K) be isomorphic to GL(m,K) if m n ? How about GL(n,K) and GL(m,L) for different fields ?
Are two compact convex sets with nonempty interior homeomorphic in R^n ?
Does a group with order divisible by d have a subgroup of order d ? What conditions can be put in order for this to be the case ?
Let G be a finite group acting on a finitely generated K-algebra R. Is the algebra R^G of invariants finitely generated ? What if the group G is infinite ?
Is a subalgebra of a finitely generated algebra finitely generated ?

>working with subhuman dog-eaters
I'll pass.

What's wrong with eating dogs?

You don't actually need to work through a whole textbook, usually when trying to learn a new subject if there's enough overlap then you can just skip ahead. In other cases people may only be interested in certain results and so they only look at those. I don't know many people who completely read a textbook since they usually don't need to, at least not in one sitting. Plus all books aren't uniformly great, for example, a lot of people who like Rudin will admit the later chapters on measure theory are just straight up bad. Dummit and Foote while being way too padded out still has some of the best exercises for the subject. I don't think I've ever seen a 'complete' PDE book, point is each book has it's own strengths and weaknesses and you shouldn't limit yourself to one nor exhaust yourself trying to finish.
The field of measure theory was started by asking pretty basic questions about rigorously defining volume and such. Manifolds was developed to understand more abstract spaces. Functional analysis is a pretty straight forward generalization of linear algebra, and complex analysis is so rigid that many of the theorems fall out naturally. I'd suggest looking at some basic definitions and results in the fields and try re-deriving them, most of the elementary results aren't very difficult to prove.

What are amplitudehedrons, /mg/?

Not him, but those are pretty fucking nice questions to ask oneself. I think being able to wonder about these kinds of things is a very important ability. Have any tips for a brainlet?
>most of the elementary results aren't very difficult to prove.
I always found this fascinating. It took so many people so many years to develop alll this stuff and now we can expect a hardworking undergrad to rederive a good chunk of it.

>I always found this fascinating. It took so many people so many years to develop alll this stuff and now we can expect a hardworking undergrad to rederive a good chunk of it.
There's a fundamental difference. A researcher will have to try a lot of things and see which paths lead to anything interesting, etc, but the problems he has solved can then be used as problems in textbooks with the assignment to prove the claims. Instead of using his intuition and reasoning to construct a bridge over a river full of unknown creatures, the student is simply ordered to repair that bridge. Sometimes he is even given hints.

Thank you, I really liked that metaphor.

It's yours my friend. Take care.

what's a good parameterization for a diverse subset of the family of functions that map the unit square to the unit interval and integrate to unity.

Feynman diagrams in the Lagrangian Grassmannian, I think. I'm not a stringer theorist.

Densities associated with bivariate copula functions?

1. only arguments can be valid or invalid, only propositions true or false
2. yes, of course it is also true, you are just instantiating y. y is a variable. that means you can replace it with anything.

Is there a topos of all axioms?

>tfw no known polynomial bijection [math] \mathbb{Q}\times\mathbb{Q} \to \mathbb{Q}[/math]

What is the best brief introduction to Fourier Analysis, for someone who's seen most other parts of analysis at this point.

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It is well known that the the category of all axioms is an abelian category.

The fuck are you talking about?
This meme is really pissing me off.

Let A be set of odd numbers
Which axioms should I use to prove that A exists, and that 3 is in A?

1 is in A because 1 = 2x0 + 1
3 is in A because 3 = 2x1 + 1

In first line you show that 1 is odd, not that 1 is in A. Isnt there a missing step from "1 is odd" to "1 is in a set of odd numbers"?

You use
en.wikipedia.org/wiki/Axiom_schema_of_replacement
If you know that the set of integers exist then does the set of odd numbers since
[eqn] A = \{ 2 x + 1 : x \in \mathbb{Z} \} [/eqn]

A is a subset of the integers by definition of odd.

Assuming the integers are a set, you can use the axiom schema of replacement (or rather, the axiom schema of restricted comprehension, which is implied by this) to create the set [math]A:=\{x| x\in \mathbb Z\;\; \mathrm{ and }\;\; \exists y\in \mathbb Z : x=2y+1 \}[/math]. The set is non empty because in particular 1 is in the set. And 3 is in the set because 3 = 2x1+1

Ok, I see that it exists. How do you the show that [math](\exists x\in \mathbb{Z} \texttt{ s.t.} 3=2x+1) \implies ()3 \in \{2x+1:x\in \mathbb{Z}\}[/math]

are you literally retarded? I told you twice. It exists because you find it, and it turns out that x is 1

Ok, now I see, thanks.

Do you guys take notes or just go straight to exercises?

Are you asking whether people read a textbook before attempting to do the exercises? Sounds pretty dumb to do exercises without some idea of what's going on.

>Assuming the integers are a set
Which they aren't.

How much is 1+1????!??!!?

I used to take notes, but then I realized it was a waste, I never really went back to them and take a look, it was more just so that I had something to do other than listen to the lecturer. So I don't take notes and just dive into exercises, but if there's something I don't understand I write a little bit of text in a box next to it and arrow pointing to the problem explaining whatever I struggled with. That's the kind of "note" i take, but nothing from books or lecturer.

you take notes to reaffirm learned knowledge during the lecture
i have shit handwriting never re-read notes

Gonna study tonight and tomorrow from Hoffman and Kunze for my final in canonical forms of a matrix and bilinear forms.

Scientifically speaking, how good is this idea? Take in mind that I haven't touched this textbook before. Just gonna pick it up to cover these two topics.

Also we shouldn't downplay the role of notation. Once we have good notation it's often much easier to find the correct way to apply them. If you want a theorem of the type "if object X has property Y then it also has property Z" then this might be straight forward when you have the property Y and Z already defined but if you don't have that then it becomes much harder. Why is Z even a good concept what about the more "intuitive" Z+epsilon (for which the proof doesn't hold)?

Hope that made some sense.

>Z+epsilon
Cringed pretty hard.

Do people actually remember names of these things
My head hurts already when trying to pronounce the name

nice

Is that a Futurama reference?

How can I interact with dark energy/dark matter?

Learn to count in greek first. After that first hurdle, it's easier than you think.

Learn Greek numbers and those names becone trivial. If you are an Amerifat, don't even try pronouncing them, it will go wrong for sure.

well?

I am Greek, so yeah.

0.5 , 0.5 , ...
and
0.49 , 0.499 , ...
are both Cauchy rational sequences and for every rational ε>0 there exists a natural N such for all naturals m and n: if m and n are greater than N, then the m-th term of the first sequence and the n-th term of the other sequence are less than ε distance apart.

ε=p/q
|xm-xn|=1/mq/p
take N greater than q/p and it proves it.

This shows that these sequences are equivalent and represent the same real number by definition.

>1/m
meant 1/10^m
this fucks up the rest, but it's easy to fix it

X is uniformly distributed over [10, 20]
Y is uniformly distributed over [10, X]

How can i calculate E(Y)?

WHY IS IT not just (X-10)/2? Does that make (E(X)-10)/2) correct?

Why do I need to use "conditional expectation" at all?

>How can i calculate E(Y)?
E(Y)=E(E(Y|X))

Also,
>WHY IS IT not just (X-10)/2?
Because that's a random variable, not a number.
>(E(X)-10)/2) correct?
Yes.

So E(Y) ends up being 12.5? Or am I retarded

Wait a minute. The expected value of Y given X=x is (10+x)/2, not minus.
So, E(Y) = E((10+X)/2) = (10+E(X))/2.
E(X)= (10+20)/2=15
Therefore, E(Y)=(10+15)/2=12.5.
Yes it ends up being 12.5.

Caught this. Thanks very much.

As for Var(Y), this ends up being

E[Y^2] - [E(Y)]^2

what???? - (12.5^2)

can you explain how I find the expected value of Y squared?

V(Y)= E( V(Y|X) ) + V( E(Y|X) )
Eve's law (EV VE)

en.wikipedia.org/wiki/Law_of_total_variance
r.amherst.edu/apps/nhorton/Adam-Eve/

any chance of getting some pseudo-math from you...this is quite new to me still

>any chance of getting some pseudo-math from you...
What do you mean?

in programming code that isn't fully written out but is fully outlined is sometimes called pseudo code

E(Y|X) = (10+x)/2 right?

i'm not sure how to get the variance of that

and then im not sure what V(Y|X) is supposed to be either, or how to take an expected value of that

>E(Y|X) = (10+x)/2 right?
Nope. It is (10+X)/2. Capital X.
Consider E(Y|X=x). You put in an x and you get an expression depending on x only. In that expression, replace x with X; this is symbolized by E(Y|X) and it's a random variable.
Similarly for V(Y|X).

V( E(Y|X) ) = 1/4 V(X) = ....

V(Y|X=x) = ....
then replace x with X and take the expected value

i follow up until "Similarly for V(Y|X)" - I only got E(Y|X) by trying to think through the problem logically (expect y to be halfway between 10 and X) and I'm not sure how to represent V(Y|anything)

>V( E(Y|X) ) = 1/4 V(X)

I don't even understand why this is true.

>I only got E(Y|X) by trying to think through the problem logically (expect y to be halfway between 10 and X)
See here . It is (10+X)/2.
The fact that its variance is 1/4 V(X) follows from V(aX+c) = a^2V(X).

>and I'm not sure how to represent V(Y|anything)
Y|X=x is "Y given that we already know the value x of X".
This means that Y|X=x follows Uniform(10,x).
The variance of it is 1/12 (x-10)^2.
Therefore,
V(Y|X=x) = 1/12 (x-10)^2
so
V(Y|X) = 1/12 (X-10)^2 .

And now you have
E( V(Y|X) ) = E( 1/12 (X-10)^2 ) = 1/12 E(X^2 - 20X +100) = 1/12 [ E(X^2) - 20 E(X) +100 ]

To compute E(X^2) you can use this:
E(X^2) = V(X) + (E(X))^2

>universe has been around forever
>universe still hasn't reached thermodynamic equilibrium
Is the second law of thermodynamics wrong?

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Universe has not been around forever, Bam problem solved.

What caused the universe to start?

>prof rewrites literally every single proof in the entire class as a "proof by contradiction" for no reason

Can someone give me the Veeky Forums book guide pictures starting from highschool?

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>he/she posted the memelist again

how do you compute zeta function over 2k, k integer?

zeta(0) = -1/2
zeta(2k) = 0, k negative
zeta(2k) can be found using poisson summation + Euler maclaurin; or else by Parseval's theorem, and by Weierstrass factorization for zeta(2)

yes, I think I think I need zeta(4 and 6), I read a formula involving bernoulli numbers(never seen in my life) but was one page of computations involving cotangent, which was a bit random for my standards
do you have any reference for those techniques you said (or care to explain?)

en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function

What do you call this
[eqn]\left(x_1, y_1\right) + \left(x_2, y_2\right) = \left(x_1 + x_2, \sqrt{y_1^2 + y_2^2}\right)[/eqn]

Is it true that every number >11 can be expressed as a sum of a prime and semiprime?

+ is defined on [math]\mathbb{R}\times\left(0,\infty\right)[/math] btw

>What do you call this
What do you mean?

>What do you call this
What do you mean by "this"?

For [math]A,C[/math] simplicial [math]B[/math]-Algebras for some simplicial ring [math]B[/math], how do I prove [math]{\pi _n}\left( {A \otimes _\operatorname{B} ^LC} \right) \cong \operatorname{Tor} _n^\operatorname{B} \left( {A,C} \right)[/math] ?