/sci/ - Science & Math

>>11135636
where the fuck does goeddel prove that god exists?

>>11135644
>where the fuck does goeddel prove that god exists?
https://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof

>>11135676
i think you need to learn what a proof is

>>11135721
did you not read it?

>>11135721
>i think you need to learn what a proof is
We feel the same about you.

Threadly reminder to work with physicists.

>>11135726
>>11135729
I did read it!... This proof is highly unrigorous and relies on a ton of assumptions about a unrigorous concept. how can you prove god exists when you cant define god?

okay sure, you can make a proof like every "thing" exists, then you just have to prove that god is a thing. and you have to define what things are.

>>11135741
>This proof is highly unrigorous
How so?

>>11135741
>how can you prove god exists when you cant define god?
all powerful, morally perfect being. Basically what the theologians have been speculating about since the middle ages. We arent talking about a first graders understanding of god and angels here, we are talking about the actual ontological objects

>>11135750
its doing logic on nonsense. inventing axioms of positive/negative, god being the most positive, and existence in our material world being a positive.

also, fails the distinction of would be most positive and true most positive, just because a real god would be more positive than a mental god, doesnt mean the hypothetical of a real god must be real. a most positive thing does exist, and you can call it god, but real counterparts dont exist for every thought, so imagining something thats worse than god in your mind but would be better if it were real doesnt make it real

>>11135765
>lesswronger's first Godel misinterpretation

>>11135774
okay then make me even less wrong

>>11135765
>inventing axioms of positive/negative
you must have some valuation of the world anyways, so why not assign positive or negative to the nature of properties? You must do that anyways. Its just a matter of what properties you give what.
>god being the most positive
This is what god is defined o be in almost all religions though. every conceivable "desirable attribute" is attributed maximally to this being
>existence in our material world being a positive.
dosent say that, it says necessary being is a positive, as opposed to contingent being

>also, fails the distinction of would be most positive and true most positive, just because a real god would be more positive than a mental god, doesnt mean the hypothetical of a real god must be real. a most positive thing does exist, and you can call it god, but real counterparts dont exist for every thought, so imagining something thats worse than god in your mind but would be better if it were real doesnt make it real
it has nothing to do with "mental" gods, i dont even understand what youre saying here

>>11135788
mental god as in the conception of god within your mind, in relation to necessary vs contingent

god exists only in mind - contingently real
god exists everywhere - necessary

point is, the #1 greatest thing will be what it is regardless of what you think. if you dream up something like a utopia, that doesnt make it exist, so even if utopia would be #1 if it WERE necessarily real, that doesnt make it necessarily real. so god can be like a shitty knockoff utopia and you cant do shit

>>11135797
>point is, the #1 greatest thing will be what it is regardless of what you think
thats literally what you are denying though, you dont understand the argument

>>11135808
maybe indirectly. it basically seems like this

good exists, thus can be put into a hierarcy of amt of good, so theres got to be a goodest thing out of all the things. that goodest thing is god

if thats the argument its hurr durr level obvious and stupid, the only flaw is if there are infinite things it might be impossible to pinpoint the best one

>>11135814
Thats the argument from morality, which also proves an all moral being, also to point out the flaw in your reasoning about the infinite: this is why god is ominbenolvlent: given any moral valued element, god is greater in goodness. Its not only the "goodest" its the supreme-um or limit of all other elements

but back to the ontological argument, Since you seem to be using the "mind" "reality" language I assume you are looking at weaker versions of the ontological arguments (Anselms etc) which dont hold up

>>11135831
given any quantity valued element, infinity is n+1, its the limit

so god is a moral infinite, eh, sure, that could work too. i just dont get the big deal

>>11135850
>i just dont get the big deal
Its like doing math, Theology is fundamentally about reasoning with your mind alone. Thats why I posted it here, speculating about the nature of god is fun like speculating about the reals or abstract algebra etc

>>11135856
but as far as math goes, its not a groundbreaking concept. its just, X exists, has hierarchy, top of ladder is hierarchon. big woop, we already did this with the naturals.

>>11135862
Theres alot more then that you can do, You can show angels and demons, as they were classically postulated by the middle age theologians, can exist like i said earlier for example.

Its just fun creating theological theorems.

>>11135869
sure i guess, theres a little puzzle shit in there but there are cooler mathematical puzzles everywhere, and the emotional gravitas of those words is heavily diminished

>>11135862
Theres alot more then that you can do, You can show angels and demons, as they were classically postulated by the middle age theologians, can exist via formal modal logic like i said earlier for example.

Its just fun creating theological theorems, its a shame so many people think that they were just saying nonsense, because i honsetly think the some of the smartest people ever were the theologians of the middle ages and the islamic golden age.

>>11135862
That was before Mathematicians were conscious of order theory. It was a faculty of thought that wasnt fully developed in the human mind.

Has anyone learnt through Tao's Analysis? How does it compare to Rudin for an autodidact?

>>11136129
>How does it compare to Rudin for an autodidact?
Rudin is a meme.

>>11135741
>when you cant define god?
You can. Gödel does so, but it might be a unsatisfying definition.

Threadly reminder to read Gromov's new popmath book
"Great Circles of Mystery"

>>11136359

i like PoMA because it actually explains basic topology before it goes into topologically-based epislon-delta-type arguments.

>>11136512
What it is about, anon?

>>11136512
>it's another "old fart rambles about consciousness" episode

>>11135990

Stats exam tomorrow /sci/ what should I look out for? I fucking hate MLE's.

>>11136566
Cars and domestic accidents. They tend to be common causes of death.

>>11136578
The stats check out, thanks!

>trying to teach myself precalculus and then calculus
>they throw questions at you that they didnt show you how to do earlier in the book
what is this shit

What's some recommended/required reading to acquire the undergrad math knowledge needed for quantitative analysis in finance?
I'm a CS undergrad currently regretting not choosing a maths or physics degree.

>>11137060
At one point you are supposed to learn how to think. You know, if the real world only had problems that were already shown in some book then there'd be no point to science or math.

>>11137060
Kinda weird but you need to know that concept. You subtract the top function by the bottom function, just algebra. Keep in mind the top function for the first one is y=3

>>11137112
Ah okay, makes sense. So
[math]f(x) = x^2 - 4x + 3)[/math]
and
[math]f(x) -x^2 +2x[/math]

Thanks

>>11137112
Ah okay, makes sense. So
[math]f(x)=x^2 −4x + 3)[/math]
and
[math]f(x) = −x^2 + 2x[/math]

Thanks

>>11137105
Quantitative analysis in finance is actually not that deep unlike most people believe, you just need to be insanely clever to notice stuff. In terms of technical requirements, all you need to know is probability theory all the way from knowing how to compute odds on a dice to stochastic calculus tops. A book that will literally take you through that whole ride is "Probability and random processes".

But if you practice quant finance then you will soon find out that anything beyond that (like what you may learn in a Ph.D.) is absolutely useless. It may even be a bit overkill as most quants will just answer questions like

"If I buy SPY at 3 PM today and sell it at 6 PM today, what is the probability that I'll make money"? This question may seem complicated, but we have years of data regarding SPY, so you could literally run this "strategy" over the historic values of the SPY at 3 PM and 6 PM and doing some basic arithmetic find your expected return (probably negative after fees). The higher stuff like stochastic calculus and the numerical methods that go with it will only be useful if you want to price instruments which is another side of quant finance.

>>11136563
I guess not. Very good. There will be some quality among the participants then.

What things can you do in number theory that aren't just jerking off with primes

>tfw qt3.14 gf catches me peaking at back of book for answers and scolds me
is there a worse feeling?

>>11136359
What about Tao?

>>11137264
>is there a worse feeling?
Not having a gf.
Never having had a gf.
Never having had a gf and never will have a gf.

>>11137139
Thank you this is very useful information.

The way quants are presented they seem like mathematical wizards, but I guess that must be in the perspective of normies, and especially business majors.

set theory exam tomorrow bros...i don't think i'll make it

>>11137394
Stop talking about me online!

What mathematical concepts would you teach all laypeople?

>>11137605
methods of proof and logical thinking

>>11137260
Use arithmetic schemes to pretend you're doing something geometric.

>>11135733
>>11135733
Love the detail on this!

Also should I drop Alg Geo? The proof of this functor on projective schemes being representable is doing my head in.

Is pic related a good book for someone who wants to practice proofs and problem solving?

>>11137605
Elementary probability theory, logic

>>11137260
A completely generic layperson only needs to know arithmetic. But children should be taught rudimentary algebra as well, because many 7th graders don't know what they want to do yet and you don't want to cuck them out of the choice of a technical career later on.

>>11137640
Can't you just develop an intuition for why it's representable and boldly skip the proof?
Representability is usually easy to grasp.

>>11137652
>Is pic related a good book for someone who wants to practice proofs and problem solving?
Why don't you read it and find out?

>>11137657
The lecturer said the proof was important to grt a good grasp of schemes. Also he hardly proves things, he just keeps referring to Hartshorne. Hardest course I‘ve ever done by far. I just wanna do number theory desu

>>11137652
It does have some fun problems in it, and given that it's a cheap Dover book (or free, if you're just going to get it from libgen) you may as well try it and see if you can have some fun with it.
But if you're not already fairly comfortable with writing proofs many of the questions in there will probably be out of your reach to solve. It's a compendium of problems from Soviet math competitions, so they're roughly intended to be something that might take the top high school students in Moscow 20-30 minutes to do.

I‘m doing Part III at Cambridge. AMA

>>11137682
Are you sure he wasn't joking?

>>11137720
He‘s so dry I would be very surprised. The proof does seem useful. Took us an hour.

>>11137682
>he just keeps referring to Hartshorne
If this means you're actually going to Hartshorne to try and understand the proofs, this is probably a large component of why you're struggling. Hartshorne is dogshit. Get ahold of some different books. You will probably need to use several sources to totally replace Hartshorne (since the second volume of Gortz/Wedhorn has been "in progress" for 10 years) but most courses don't do the whole book anyway and each piece will be much, much easier to understand than the unmotivated half-skeleton of a proof in Hartshorne.

>>11135676
>Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza
pffffffffffffffffffffffffffffffffffffffffffffffffffft

>>11137640
dropping schemes? t.t
what functor are you talking about?

>>11137924
It assigns a scheme T over a base scheme k the set of surjections from the n+1-fold direct sum of the structure sheaf of T to line bundles over T modulo an equivalence relation. Apparently it‘s represented by projective n-space.
I know I‘ll need schemes eventually but I‘m tempted to take an easier exam than this one.

Guys where should I start with math as a brainlet? ^_^

What order should I learn things in %_%

>>11137949
Look up a good university study plan, see what they do and their bibliography, go on from there

>>11137937
O MY THEOREMS ABOUT LINE BUNDLES ON PROJECTIVE SCHEMES
Anyhow, the n+1 fold product is the set of regular, locally defined maps to A^n+1, and A^n+1 is the tautological bundle over P^n.

>>11137937
i'm not so sure that it is appropriate, but iirc there was a chapter on Gorz giving the reasonable condition for presheaf to be representable by a scheme, and the main motivation was grassmanians (your example in a less fancy language)

>>11137949
What is your math background?

>>11137996
high school level but i am bad at using algebra tricks to solve exercises, i don't know how to even learn the properties of logarithms, exponentials, etc.

>>11137937
>>11137987
> I know I‘ll need schemes eventually but I‘m tempted to take an easier exam than this one
that sounds a tough course

>>11137987
Thanks!

>>11138024
Sure is. Halfway there though.

>>11137638
Like algebraic geometry?

my question: there is this formalism developed by Elie Cartan on orthogonal coframes
i feel very comfortable with differential forms and like those formulas very much
there is a book by Cartan recently (2002?) translated into english, which was a nice reading
this formalism was also briefly mentioned in an exercise in Petersen (but not much was said afterwards)
!! unfortunately the story ended too soon
can you recommend a book on riemmanian geometry (/surfaces/whatever) using this formalism?
i want to compute things like laplacian and friends, avoiding Koszul approach
(no clelland, no cartan's methods in homogeneous spaces)

>>11138067
*those thetas are meant to be omegas

Does anyone have a recommended text or lecture series for learning sheaf theory? I shouldn't be missing any of the building blocks they use for examples, though I never really delved into algebraic geometry, so I could be horribly wrong.

>>11138134
asking but not going to answer, do you know the context where you want to use them?
- complex geometry (riemann surfaces) / complex analysis
- algebraic topology
- algebraic geometry
- ??

(ok, i'll wisper the name of a book but it could be not suitable, Taylor's several complex variables)

[eqn]\lim\limits_{n\to\infty}\mathbb{Z}/n\mathbb{Z}=\mathbb{Z}\text{ or }\{0\}?[/eqn]

>>11138161
Algebraic topology is my drug of choice. I don't care to learn algebraic geometry unless it's essential.

>>11138164
I think the limit was one and the colimit was the other one, but I can't recall.
>>11138173
Glen Brendon.

>>11138200
What I do remember is the following:
Z is the smallest group having at least one non-trivial map to each [math]Z_n[/math].

>>11138216
so a that would be the limit
but lim with arrow to the right is colimit
(recall that arrows go from top to bottom
and from left to right)
so that's not what you want to compute,
assuming it is written correcly

also you should specify the category
(ab, rng, ring, ..)

>>11138008
i would start with the firt 2-4 chapters of spivak's calculus, it covers elementary algebra, graphs and the other high school stuff

>>11137394
i know, i lied. it was actually a blissfully comfy feeling

>>11137605
basic statistics and related topics are probably the most impactful thing for the masses to understand

>>11138200
One of the few books from that series I can't find in pdf form, curses.

>>11137949
>What order should I learn things in %_%

>>11138423
I'll get past high school one of these years

brainlet here again
first x2 +1 = 3x
then x2 = 2x
now im fucked. square root of 2 is not x. How can I solve for my low IQ

>>11138884
I don't even understand the answer... should I just kill myself

>>11138889
>>11138884
First let's assume x!=0. This holds as a direct substitution shows that x=0 is not a root.
Let's divide by x. This yields
x-3+1/x = 0
Therefore, x+1/x=3 - the answer is d.

Oh, and whoever types fractions like this is a massive faggot.

>>11138200
dumb weeaboo

>>11138919
So a divide stage removes ^2?
so then you added 3 for -3, so the other side must now but +3 ok
>x +1/x =3
3 + 1 / 3 = 1.33333?

>>11138928
>So a divide stage removes ^2?
Are you familiar with the definition of ^2 or just retarded?
>3 + 1 / 3 = 1.33333?
You substitute x=3 in an expression that is valid only for a root of the initial equation. Is x=3 a root of the initial equation?

>>11138962
Im retarded, sorry
double retarded, you have to forgive me
when I test 3
3^2, 9 and -3 times x(9) to -9, + 1 = 1?

Pi is 22/7.

>>11138971
Yes, since 1!=0 this means x=3 is not a root of the initial equation.
Just do more exercises, grab a book where solutions are explained (I am afraid you have skipped on themes earlier than quadratic equations though), and you'll gain better understanding.

>>11135833
So does the textbooks listed here in actually reside in any single trove or archive or is it like a go find it yourself brainlet sorta deal?

>>11138979
Yeah I'm going to go through more khan academy I guess. thanks for your assistance

>>11137060
literally just think for more than 2 fucking seconds. christ.

>>11137345
no one cares about your experience with elementary children's analysis books. go bother someone else, freshman.

>>11138993
Tao's book is trash but no need to be rude. This is /mg/, go be an asshole to newbies in the cesspool you came from.

>>11139001
this is the place where we're assholes to newbies. people are nice to newbies in /sqt/ and on reddit. fuck off tourist.

>>11138164
Nope, neither.
The limit is the so-called profinite completion of Z, the colimit is [math]\mathbb Q/\mathbb Z[/math] (which you may think of as the group of all roots of unity)

>>11139114
What degree do you have?

>>11137605
Addition and multiplication.

>>11139189
I have an MS, pursuing a PhD atm

>>11137605
Depends on the goal.
The most useful is probably basic logic and stats.
For fun, maybe some discrete math or topology, anything you can draw. Probability is good for this too

guys waht if u cant solve p=np or raymann's hypnosys cuz of girdle incomplete problem? any1 thout about that???

>>11136512
I'm convinced that Gromov "sees" stuff in his head that other people will never be able to even dream of.

>>11138200
>>11138259
>>11139114
Damn I was just joking around with that, didn't know there could actually be a meaningful answer.
Thanks anons.

What math do I need to start learning ML?

>>11138392
Doesn't .djvu work?
>>11139794
I mixed it up, please don't thank me.
>>11139799
Linear Algebra and optimization.

>>11139801
>linear algebra and optimization
No statistics?

Are there any ways to keep up with or go in depth into a field without knowing and talking to people who have done so?
Specifically, how do you find out about nice papers?

>>11138884
>>11138889
>>11138928
>>11138971
Note that you are NOT trying to find x here (it happens to be (3±√5)/2, but that's not relevant).
> x^2-3x+1=0
Divide each term by x:
=> x-3+1/x=0
Add 3 to both sides:
=> x+1/x=3
Now you have the answer.

>>11139801
Did not know of this file type. Thanks anon; I now have access to many new resources. Also Bredon seems to be written to my level, so thanks again.

>>11139014
you actually tend to put peanut butter on your crotch for the newfags from what I've observed, this board is also the closest to reddit of the pseud boards

Anyone know a good place to get pdf or djvu files of math or physics books? The place I was using doesn't want me grabbing more than 5 a day. Yes, I horde.

>>11142005
Libgen?

>>11139799
Basic Analysis, linear algebra and numerics skills.

>>11142233
Links are always dead for me even if they "have" the book I want.

>>11140948
have you considered that you don't belong here? have you even taken a moment to ponder on the possibility that you aren't capable of making it in higher level mathematics?

>>11135638
Brainlet question but here it is:
If
[math]u_{xy}=u{yx}[/math]
does it follow that mixed partials in x and y of all orders commute?
For example, does it then follow that
[math]u_{xyy}=u_{yyx}[/math]

>>11142330
First line should obviously read
[math]u_{xy}=u_{yx}[/math]

>>11142330
I don't know the typesetting on here but:
let v = u_y
u_yyx = d/dy v_x = d/dx v_y = u_xyy

Assuming all derivatives involve exist and are continuous.

>>11142352
But why should partial derivatives commute with
[math]v=u_y[/math]
just because they commute with [math]u[/math]? Does this step not require some justification, or does it follow immediately if mixed partials of [math]u[/math] commute?

>>11142330
>does it follow that mixed partials in x and y of all orders commute?
Yes, simply by induction.

>>11142373
>But why should partial derivatives commute with
I don't get what that means.

>>11142255
Tell me the name of the book and I will have a look.

>>11139014
Who is we ? Fuck you and your inferiority complex

>>11139001
Why do you think Tao's Analysis is trash?

>>11137060
If you find a problem hard that means you're learning something. If you only ever did easy problems you'd never grow. Keep at it.

>>11142330
This is only true if the derivatives in question are continuous, so in general you can‘t assume this.
>>11142389
Stop bullshitting

>>11142541
So equality of all higher order mixed partials follows from equality of [math]u_{xy}=u_{yx}[/math] if and only if [math]u[/math] is sufficiently smooth? Or is it strictly only if?

>>11137692
College? Which courses?
I'm currently doing part 2 and it's much harder than first 2 years desu

>>11142541
>Stop bullshitting
???

What's a good high-quality analysis textbook suitable for someone who already has a math background? I'm an ex-logician who saw Christ's light and understood that I was irrationally thinking low of proper maths. I'm already going through algebra (Atiyah-McDonald and Eisenbud) and topology (Munkres and other random papers), now I need one for analysis.

>>11142330
>does it follow that mixed partials in x and y of all orders commute?
absolutely not

>>11142809
Under what conditions, aside from the function u not being smooth enough, would that not follow?

>>11142814
continuity of partial derivatives of order k implies the symmetry up to order k. so "u being smooth enough" literally means that the derivatives commute.

>>11142819
Right, I just thought that since other people have already said [math]u[/math] is sufficiently smooth if and only if its mixed partials up to order whatever commute that he was disagreeing with some point there.

>>11142774
Baby Rudin for the basics, Cohn (for integration), Brezis

>>11142591
For 2nd order partials, you need [math]C^2[/math], so I'd guess for each higher order bump up to [math]C^3[/math] and so on.

>>11142827
don't worry, you're talking to a bunch of autists who can't seem to answer your original question.
why do single partials commute in the first place? try writing down the definitions and seeing if you can "swap the places" of x and y. i.e., writing down the difference quotient lim_(t to y) (u_x(x, y) - u_x(x, t))/(y - t), then write the u_x's inside as their own difference quotient. simplify by moving everything into the inner limit, and then you have to swap the two limits - think about why this makes sense by continuity of second partials of u. finally, undo your simplifications to swap the places of x and y, (and your limiting variables) and rewrite as u_yx.

>>11142774
The stein and shakarchi series, if you really want to dig in. If not, the first volume has more than enough analysis and basic functional analysis.
But you'll probably want a bit of their complex and fourier books as well.

Sophomore in undergrad here

Is taking abstract algebra and measure theory in the same semester a bad idea? I just signed up for them both, but I'm worried that I may be overwhelmed.

>>11142373
>>11142814
The justification for the manipulation is "assuming all derivatives exist and are continuous." The argument is literally an induction argument and the link below is a base case.
https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz's_theorem

>>11143162
His question WAS answered. The answer, despite being incredibly simple, was too much for him to understand.
>>11142391
Riemannian Geometry and Geometric Analysis by Jost

>>11143208
Depends on your background. Did you do well in real analysis and topology? That's the prereq for measure theory. Look up real analysis for graduate students (it is free here http://bass.math.uconn.edu/3rd.pdf).). If it looks too challenging for you, you are not ready for measure theory.

How much algebra background do you have? I was taking multiple grad classes as a sophomore and I'm not exceptional, so if you have the background, it's fine.

>>11143230
My measure theory class doesn't take topology as a prerequisite. It's called "intro to functional analysis and measure theory." Based on what you've said, I'm going to infer that it's only going to cover a few topics in the field without going into much detail.

As for algebra, I did well in linear algebra, particularly on the theoretical side, and I'm really good at proofs. I also did well in number theory, so I think I should be okay.

>>11143243
Give that book a look regardless to familiarize yourself with the content. The arguments are not all that different from those for undergrad analysis. If you could list the course syllabus I could more helpfully judge the challenge.

AA is much easier than measure theory, even at the grad level, in my opinion. If it's the undergrad AA, it will be a cakewalk. Look up groups, subgroups, and quotient groups. If you can wrap your head around these, ring theory won't be much more technical. Fields/galois theory are much easier in my experience, as what you expect to be true will be, and for the reasons you anticipate. The last item on the AA syllabus will be a generalization of linear algebra, most likely.

I've got another 2 years left, should I hold off on AA next semester so I can take the second part of intro to german? Keep in mind RA will prevent from going to the German lab.

>>11143303
It's good to get language out of the way. Understanding German to the point that you practice thinking in the language will enhance your intellect.

>>11138173
If you don't care to learn AG then you should probably stick to DG/analysis/combinatorics.
Outside of those fields all roads lead to AG.

>>11143314
I mean, it'll be the third after English (mother tongue) and French, so it's mostly for fun

>>11143354
German will be easy for you, then. English is also my native tongue; knowing it well makes German/Dutch incredibly natural.

>>11143350
Well, if it's essential, then I'm interested. None of the algebraic geometers did a compelling job of finding my interest; they always recommend Hartshorne to a newcomer.

>>11142330
This is true whenever u is a C^k function

>>11143359
The rule of thumb I've found is that "the farther you think you are from AG the more terrible the objects you'll be obliged to think about once you realize you need AG".
For example: If you do complex geometry you'll mostly think about smooth projective finite dimensional objects.
By contrast much of algebraic topology (read: stable homotopy theory) is controlled by pro-Artin stacks (c.f. COCTALOS).

If you're willing to give more specifics about what you're interested in and why I can probably make some recommendations.

>>11143374
My current interests are primarily of the physics pleb type: lie groups acting on 4-manifolds, bundles, and cohomology theories. When I was a physics student, sheaves kept showing up in my research, so I decided to see what I'm missing. My interest is almost entirely curiosity, so babby's first algebraic geometry is what suits me (already found the bredon book some other anon suggested).

My pure math interests are more of the k-theory flavor (already have plenty of exposure to classical algebraic k theory, MIlnor k-theory, topological k theory, and k theory for C* algebras). Don't know if that helps you recommend, but there's no harm in saying it, I guess.

It's well-known that given an absolutely convergent series of reals, any permutation on the order of summands will not alter its sum, i.e. the permutation series will still converge to the same limit. ([math]\star[/math])

Let [math]\sum_{n=1}^{\infty} x_n[/math] be absolutely convergent. Let [math]\{N_1 ... N_k\}[/math] be a finite partition of [math]\mathbb N[/math] *such that each [math]N_i[/math] is infinite*.
(1.) Does it necessarily follow that [math]\sum_{n=1}^{\infty} x_n = \sum_{i=1}^{k} \sum_{n \in N_i} x_n[/math]?
(2.) If so, does the same apply for any infinite partition of [math]\mathbb N[/math] to infinite subsets (i.e. letting [math]k=\infty[/math])?

As far as I can see, ([math]\star[/math]) only applies to permutations on [math]\mathbb N[/math] - and one simply can't define a permutation that would order *all* elements of some infinite proper subset of [math]\mathbb N[/math] before all other naturals. (Permutations can scramble the order on the naturals freely, but they can't "put all the even numbers before the odd numbers". This is of course easy to prove.) So even if the above equality holds, I don't see why it should follow from ([math]\star[/math]).

>>11143392
you guys are such homos lol

>>11143407
How can you have a finite partition of an infinite set which is such that each element of the partition is finite?

My instinct is to go back to definitions and look at the sequences of partial sums. For a given natural number k, there is a natural number m, so that if n >=m, the two sums have at least k terms in common. This isn't a proof, but that's my instinctive route.

>>11143472
You also chose to participate in the thread full of homos, faggot.

>>11142301
I have no interest in mathematics aside from what's useful in the sciences, the thought of wasting my life playing with myself intellectually is disgusting to me.

>>11143479
Do you not like games or puzzles? Cause that's what most who like it seem to treat it as.

>>11143485
>games and puzzles
when bored of course, everyone likes games and puzzles when they have nothing better to do anon.

Frens this question drains the life out of me and I can't make sense of it. I am only sure of the base case and the rest of the "solution" is most likely just BS.
Link on pic related: bitdotlyslash2WtO2l5

>>11143501
Work I did on question. Help is very much appreciated.

>>11143475
To clarify, what I meant was a finite collection of pairwise disjoint proper subsets of [math]\mathbb N[/math], such that each of those subsets are infinite.
For example, letting [math]N_1[/math] be the set of all odd numbers and [math]N_2[/math] be the set of all even numbers, then [math]\{N_1, N_2\}[/math] constitutes as such a partition.

It's also possible to partition the natural numbers to (countably) infinitely many pairwise disjoint infinite subsets of [math]\mathbb N[/math].

>>11143506
You directly equated the sum from 1 to k with the sum from 1 to k+1.

>>11143519
That's the thing. After base case I just messed up. I don't even get what i is supposed to be.

>>11143407
This is true (even with an infinite partition) and it's not that bad to show if you're allowed to use sledgehammers, since it follows from the measure-theoretic theorem that integration (over N with the counting measure, in this case) is countably additive.
So you must be able to prove it by throwing enough epsilons at it too. The finite case doesn't seem to be that hard, but the infinite case looks like a gigantic pain in the ass unless I'm just dumb (also possible).

>>11143577
Thanks anon, I'll take your word for it since I know next to nothing about measure theory (taking calc 3 at the moment). Good to know that the theorem holds even if I can't prove it using my current set of skills.

(Of course if you/anyone else has some proof sketch that doesn't rely on measure theoretic sledgehammers, I'd be interested.)

>>11143631
The finite case would go something like this, I think.
Assume you have a k-subset partition. Since subseries of A.C. series are A.C., the sum over each part exists; say the i-th sum is L_i. Take a partial sum of each part large enough that every partial sum is within e/k of L_i.
Now rearrange the parent series so that all the partial sums are sitting at the front (this is okay, since there's only finitely many terms being rearranged). Your series will be something like L_1+L_2+...+L_k+e+(leftover stuff). Then you just need to show you can get the leftovers arbitrarily small, which you can do by pointing out that if you shrink e enough, your leftovers will begin arbitrarily far out in the series.

The problem is that very little of this makes any sense for infinite partitions, obviously.

I'm doing Spivak's Calculus, and something he did on chapter 1 baffled me
>|1 + sqrt2 - sqrt10| = sqrt10 - sqrt2 -1
Why did he invert all the signals? Because that sum would be negative?

>>11143475
>which is such that each element of the partition is finite?
He wrote infinite at the end, not finite

>>11143775
Your explanation makes sense

>>11143852
Thanks? I just found it weird because I always heard of modulus as a way to "positivize" numbers under any given circumstances.

>>11143392
From what you've said I'd suggest that it's not worth really buckling down to learn about sheaves/topoi. For DG applications sheaves are just some extra words for keeping track of how various objects transform under coordinate changes. Essentially you should be able to pick up what you need as you go.

Given your interest in both physics stuff and K-theory I'd suggest you learn about the Atiyah-Singer index theorem and its various generalizations. Various people have suggested that there should be an analog where you do index theory on the free loop space of your manifold and replace K-theory will elliptic cohomology. Although this is certainly fertile ground it has a (well earned) reputation of being incredibly difficult to prove anything in this vein.

Something else that came up in Spivak right after, if you'll allow me:
(|a+b|)2 = (a + b)2 = a2 + 2ab + b2
<=a2+2|a|.|b|+b2 (why did he suddenly add a '<=' ? Previously he was talking about how |a+b|<=a-b IF a>=0 and b<=0, but there's no specification as to a and b's values in this equation with the square value, and he goes back to equals next - moving on...)
=|a|2+2|a|.|b|+|b|2
= (|a| + |b|)2
Can you help me understand this, please? On top of my remarks above, why did he suddenly used |a|.|b| on the second line if he theoretically was handling the (a+b)2 and not the (|a+b|)2 anymore?

And looking at the exercises for Spivak's it seems like what I imagine a book of proofs would be like.
>prove that if x2=y2 then x=y or x=-y

Is game theory a meme or a field actually worth studying?

>>11142774
Amen.
>>11143392
Jesus fucking Christ, the strength of my disgust is immesureable.

>>11142255
You’re stupid.
>>11144288
shut the fuck up retard

>>11144498
Good, seethe more.

>>11144511
Get that broom out of your asshole, cunt.

whats the use of taylor series

im gay lol

Any good intro resources on Persistent Homology? I understand basic homology but know nothing about persistence.

>>11144605
Approximating things. For example your calculator might use a Taylor series to calculate sine/cosine values using only addition/multiplication.

>>11135875
I truly pity you

>>11143243
did you do compactness and proper metric spaces in real analysis? cause if not you're absolutely fucked for measure theory.
if you did, you're only mildly fucked.
topology should really be a hard prereq for any measure theory or functional analysis anywhere.

How many books have set theory as first chapter?

>>11145140
At least 1

>>11145140
Some have it as the 0th chapter.

>>11135875

>>11145140
ive never seen an undergrad analysis text that did not have set theory in the first chapter

>>11144605
As well as >>11144755, Taylor series allows you to extend the domain of a function to other types, e.g. complex numbers or matrices. If addition, multiplication of elements and multiplication by a scalar are defined, then you can evaluate any polynomial and thus any function for which the Taylor series converges.

>>11143211
>Riemannian Geometry and Geometric Analysis by Jost
Works without issue. Click on the first mirror and then GET.

>>11145073
>topology should really be a hard prereq for any measure theory or functional analysis anywhere.
But generally topology is just retarded, a lot of real analysis (like PDEs) just use normed spaces and if you get to the point where really abstract topological results are meaningful then you really just deserve the suffering.

Every fucking grad under grad has fucking set theory first chapter .I am tilted. Ya buddy I got the idea of union, intersection, modularity, various fucking notations to represent same set, Cartesian product, power fucking set, exponential set. Just fucking end it buddy

>>11144293
It's is practical thing you should understand for your day to day living. Just like Utility theory and decision theory .

is a BA in math (double major with computer >science) a meme degree?

>>11135833
Is there a list before this for pure morons trying to even comprehend a math? I need a list of books to comprehend basic maths, numbers into other numbers, triangles in 3D, the basics. Then I'll be ready for K-theory as a cohomology factor. Where's the primer of the primer list

>>11145889

>>11145715
>ok now lets talk about locally convex topological vector spaces
>oh, you don't know what a topology is? uh.... i guess we'll talk about neighborhoods determined by a family of seminorms then...?
>hey guys lets talk about the dual space with the weak* topology
>what is the weak* topology??? uh... it's.... umm....
you're braindead

>>11146026
>>ok now lets talk about locally convex topological vector spaces
Which subvectorspace of L^p is that?

>>oh, you don't know what a topology is?
It's the set of open subsets of a set.

>>what is the weak* topology??? uh... it's.... umm...
I literally do not know what it is. You show things via weak convergence anyway.

>>11145938
thank ya chief

>>11145763
If you're not autistic, you realize that nobody is supposed to read chapter 0 and it's just there so the authors can claim their book is "self-contained".

Friendly reminder that Terence Tao is a fucking FRAUD

>>11146348
lel. Which identity?

>>11146353
https://arxiv.org/pdf/1908.03795.pdf

Eigenvectors from eigenvalues

>>11145763
What sort of books are you fucking reading?
I haven't seen set theory in the beginning of a book for years now.
Differential forms are pretty recurrent, tho, but having Cartan's identities on the book is honestly really convenient.

>>11146359
Did he think there's some 2 page low hanging fruits in linear algebra to be published, still?

Well anyway, I suppose if it brings attention to the theorem, then why not.
I don't like Tao since he uses his blog for political propaganda, but he's probably not a bad person.

>>11135833
In an ideal world... Sigh. Amerifat here. Our math is abyssmal. I work as a tutor and I mostly see 40 year old women wanting to become a nurse fuming that they can't factor properly and how useless it is.

Hey, I wanna ask you guys something.

So we have, say, addition (+) and multiplication (*).
Let's say 0 is "addition's zero" since a+0=0+a=a, and 1 is "multiplication's zero" since a*1=1*a=a.

Now, if we iteratively add to "addition's zero" one "multiplication's zero" at a time, we'll end up with a natural number sequence: 0, 1, 2, 3, 4, 5, 6,..
If, however, we iteratively multiply "multiplication's zero" onto one "addition's zero" at a time, we will end up with this sequence: 1, 0, 0, 0, 0, 0, 0,..
One sequence doesn't have repeating elements, while another does, what gives?!

>Spivak coined Spivak pronouns, a set of English gender-neutral pronouns.[7]
I should learn not to read authors' wikis.

>>11146054
no problem. if you want more detailed info about one of the subjects, pick up a school textbook. it's mostly examples and fluff, but will break down things decently well

>>11147034
this made me sad as well

>>11146949
>One sequence doesn't have repeating elements, while another does, what gives?!
Why do you expect the two sequences to behave the same? Addition and multiplication are fundamentally different.
Your question basically boils down to asking why multiplication by by the additive identity is the additive identity. One reason for that is that otherwise we wouldn't have commutativity.
It is worth noting that there are rings (sets with addition and multiplication) so that your first sequence repeats. For example in [math] \mathbb{F}_2[/math] the first sequence is 0 1 0 1 0 1 0 1 ...

[math]f(x)^2 + g(x)^2 = (g \circ f)(x)^2[/math]
?

>>11147108
what did he mean with that notation

>>11147034
>>11147055
wtf is this abomination
https://en.wikipedia.org/wiki/Spivak_pronoun

>>11147102
>Why do you expect the two sequences to behave the same?
Well, I basically did my damnedest to formulate the problem in symmetrical terms, getting asymmetry as a result. The only source that asymmetry could stem from, I could think of, is distributivity - but I currently fail to see, how exactly.

>>11147108
f(x) = g(x) = x
x^2 + x^2 = x^2

>>11147156
[math] 0 \cdot 1 = (1-1) \cdot 1 = 1 \cdot 1 - 1 \cdot 1 = 1-1 = 0 [/math]

>>11147156
You can state distributivity as follows.

Given [math]A[/math] a finite sequence of numbers of the kind you like.
[eqn]k \cdot \sum_i A_i = \sum_i k \cdot A_i[/eqn]
i.e., you can just multiply each member of the sequence by [math]k[/math]

From this and the fact that summing over the empty sequence [math]\varnothing[/math] equals 0, you get that 0 absorbs multiplication, since the empty sequence has no members to multiply [math]k[/math] with.
[eqn]\sum \varnothing = 0[/eqn]
[eqn]k \cdot 0 = k \cdot \sum \varnothing = \sum_i k \cdot \varnothing_i = \sum \varnothing = 0[/eqn]


This fails.

[eqn]k + \prod_i A_i \neq \prod_i k + A_i[/eqn]
[eqn]k + \prod \varnothing = k + 1 \neq \prod_i k + \varnothing_i = \prod \varnothing = 1[/eqn]

There's the asymmetry.

>>11146502
>>11147055
Brainlets seething because accomplished mathematicians have non braind dead views on society and politics.

>>11147224
Having subtraction isn't part of the initial conditions though (only addition and multiplication are). Is there a way to make do without it?

>>11147034
>>11147055
lol you nazi fucks gender traditionally was just a concept in language, many have three genders or none. When you learn things like Spanish you have to remember random shit that computers are feminine. Never had much to do with sex until recently.

>>11135788
>This is what god is defined o be in almost all religions though. every conceivable "desirable attribute" is attributed maximally to this being
Replace good by evil and it proves the existence of the devil.
Now, which is more powerful, God or the devil? God would be more good if he was stronger than the devil and the devil more evil if he was stronger than god.
A contradiction.

>>11147248
I think you need at least a cancelation property
[math] 0\cdot a = (0+0) \cdot a = 0 \cdot a + 0 \cdot a [/math]

>>11147248
I am sorry, I didn't correctly attack the problem. First I should determine the minimal possible set of axioms (compatible, however, with "usual" understanding of "addition" and "multiplication" however) under which these two specific statements
>Now, if we iteratively add to "addition's zero" one "multiplication's zero" at a time, we'll end up with a natural number sequence: 0, 1, 2, 3, 4, 5, 6,..
>If, however, we iteratively multiply "multiplication's zero" onto one "addition's zero" at a time, we will end up with this sequence: 1, 0, 0, 0, 0, 0, 0,..
Then, AFTER determining that minimal set of axioms over addition and multiplication - THEN should I start asking stupid question about why no symmetry.

>>11147241
I appreciate your effort, and I am sorry I have confused you with my badly formulated problems. I specifically want to make do with minimal possible instruments - and yet I haven't determined what they even are. I am sorry for wasting your time.

>>11147290
>First I should determine the minimal possible set of axioms (compatible, however, with "usual" understanding of "addition" and "multiplication" however) under which these two specific statements
>>
>>
>hold
Fix.

>>11147273
grammatical gender could of just as well been called 'color' in that it only exists to classify nouns for use in declensions. damn PC police have to ruin everything though.

>>11137937
This is basically just the definition of projective space.

Is there anything "beyond" the Church-Kleene ordinal, [math]\omega_{1}^{\text{CK}}[/math]? I can't seem to cope with how fucking big this thing is.

>If there is a model of ZFC, then there is a pointwise definable model of ZFC. Indeed, there are continuum many non-isomorphic such models.

>>11144293
I don't know. Nash's son works at the pizza joint near me. I'll ask him what he thinks. He was a combinatorialist I believe.

>>11147108
kek

>>11147526
Please be Papa John's.

>>11147359
>Is there anything "beyond" the Church
Extra ecclesiam nulla salus.

>>11146036
good lord. you have no clue what you're talking about. come back when you're above the age of 15.

>>11148488
Not an argument.
Stop pretending dude.

>>11147290
>I appreciate your effort, and I am sorry I have confused you with my badly formulated problems. I specifically want to make do with minimal possible instruments - and yet I haven't determined what they even are. I am sorry for wasting your time.
That's (possibly) the minimal answer. Using Sigma and Pi notation (justified because addition and multiplication are monoids) is the fastest way to state distributivity in a way that makes it clear that it really is the culprit, by forcing 0 to absorb multiplication because it must distribute over the empty sum as well.
It's not about addition and multiplication either. As soon as a binary operation (associative and commutative) with unit, which you call multiplication, distributes over another binary operation with unit, called addition, the latter's unit becomes the former's zero, so the second sequence will always be unit, zero repeating. The other way around, the sequence repeats iff repeatedly adding the unit leads to a cycle.

I'm pushing you towards the "minimal set of axioms" you need and a better formulation. I don't personally think it's a very worthwhile question, but questions with little value should have a trivial answer rather than none at all so there you have it.

I've been trying to learn category theory (currently ready Seven Sketches on Compositionality) because I think it might have some insights into what are normally thought of as pure graph theory problems, in particular web-of-trust like networks like PGP.

I'm thinking that a particular subgraph of the global trust network (the network of connections known to any one particular person) can be modeled as a preorder, and any time you talk to anyone else on the network there is a functor that represents the connections you're willing to reveal to that person so that they trust you enough to agree to whatever the operation is (forwarding packets for you on a Tor-like network, sending you buttcoins, whatever).

The upshot of this is that in your normal interactions with this network, you can reveal aspects of your identity without fully revealing who you are and get by just fine, and construct ephemeral abstract identities that do exactly one thing (e.g. "someone in New York who likes Italian food" as opposed to "John Smith who lives at apartment 123..."), which provides a nice middle ground between tech companies controlling everything and being able to trust literally no one like in Bitcoin or Tor.

I was thinking concepts like adjoints, limits/colimits, etc. could help with expressing some of these complex relationships so that the nodes on the network have a common language to say exactly how much information they need to trust someone else and no more, or am I way overthinking this and should just study graph theory?

How can x2y^(n-2) and -(y2x^(n-2)) cancel each other out? Can you say they're 'opposites' in some way(their exponentials switch)?

I couldn't format the n-2 in latex, but I think it's understandable like this.

>>11148890
Do they ? Aside from the fact that one is equal to the other with the arguments flipped (so that they are opposites if x=y), I don't see why there should be any connection between the two

>>11148890
>How can x2y^(n-2) and -(y2x^(n-2)) cancel each other out?
n=3, else they do not, at least if I understand your writing (n=4 if by x2 you mean x^2).

>>11148953
I think that would be the case in this example, it's a fragment from a series from Spivak's calculus, chapter 1 exercises. The question is "prove that x^n-y^n=(x-y)(x^(n-1) + x^(n-2)y + ... + xy^(n-2) + y^(n-1) )
I thought just expanding it wouldn't do much(multiplying x by the following expression and then do the same with -y), but that's just what the solution does, and it suggests you'll arrive at the result by cancelling values within the series. xy^(n-1) and -xy^(n-1) cancel each other neatly, for example, but at the end there's still the x2y^(n-2) and -(y2x^(n-2))

>>11148985
From the solution

>>11148697
>or am I way overthinking this and should just study graph theory?
You're massively overcomplicating simple things to try and hamfist them into a category-theoretic framework. Just study graph theory, and as a bonus tip, coming up with grand ideas before you know what you're talking about almost always turns out poorly.

>>11148985
>>11148990
>but at the end there's still the x2y^(n-2) and -(y2x^(n-2))
No, there is isn't.
Look at the solution again and remember that the ... stand for the other terms, everything except the x^n and -y^n cancels.

If you have trouble understanding the last simplification write out the entire "..." for n=4 and you will see it.

>>11148990
You are monkey.
Monkey no ask question in /mg/, okay?

>>11148990
>spivak
oh no...

>>11149142
I know what the ... means, but after writing it out and not just trying to do it algebraically I'm obviously doing something very wrong.

>>11149170
What's the matter?

Stony Brook University or New York University? I don't know which to choose.

/sci/, I cannot for the life of me finish the proof that for any non-empty closed [math]F \subset \mathbb R^n[/math] and non-empty compact [math]K \subset \mathbb R^n[/math], the distance [math]d(F,K) := \inf \{d(f,k) | f\in F, k\in K\}[/math] is achieved. ([math]d[/math] is a metric on [math]\mathbb R^n[/math].)

Here's what I've got: Let [math](f_n, k_n)_{n\in \mathbb N}[/math] be a sequence such that [math]d(f_n,k_n)[/math] converges to [math]d(F,K)[/math]. By compactness, let [math](k_{n_j})_{j\in \mathbb N}[/math] be a subsequence that converges to some [math]k\in K[/math]. Given an arbitrary [math]\epsilon > 0[/math], let [math]J\in\mathbb N[/math] be large enough such that for all [math]j>J[/math] it holds (using the triangle inequality) that [math]d(f_{n_j},k) \leq d(F,K) + \epsilon[/math].

I'm not sure how to proceed. I'd like to show that there exists a subsequence [math](f_{n_{j_l}})_{l\in\mathbb N}[/math] that converges to some [math]f\in F[/math], and this would finish the proof if I'm not mistaken.

Any help is greatly appreciated!

>>11149200
For a generic undergrad I would choose NYU. Even though the two universities are probably comparable, brand name and connections are somewhat important in undergrad and NYU's name is way better. Also you'll get to spend 4 years smack in the middle of lower Manhattan instead of some little shitstain suburb.
For graduate schools it's impossible to say without knowing what you want to do. Stony Brook generally has better research faculty IMO, especially in pure areas of algebra and stuff like that, but they do get trounced in some areas (applied math being the biggest one, but NYU has a giant probability group too).

>>11135638
Whats up.

After being recommended at my unis math department by multiple profs, im starting a complementary study into math (not exactly a minor, its better than that) next semester as a EE student.
Ill start with analysis 1 which covers:
>Everything to do with R, its construction, topology, etc
>Differentiability
>Infinite series
What can I do to prepare myself for this?
What can I expect?

>>11149181
You didn't actually write the "..." notice that the exponents always sum up to n.
In your second bracket it should be (x^3 + x^2y + xy^2 +y^3)

>>11149294
Wouldn't "not writing the ..." actually be the way you just wrote(without xy and 1)? I thought of putting those so the exponent was reduced down to 0 and then back again.
But it worked with your suggestion, so thanks for all the help

>>11149288
>What can I do to prepare myself for this?
Read the course text beforehand.
>What can I expect?
The topics in the syllabus.

>>11149322
I think your confusion stems from the "..." being left out and it not being entirely clear what is meant, these are always points at which you have to consider what exactly the symbols mean.
But I am pretty sure that my suggestion is what is being asked.

>>11149332
When I ask these things Im obviously talking about the experience as a whole and not about what im going to learn specifically.
You know this tho, you're just choosing to be a prick because you think it nets you "4chan points" or someothing.
Truly sad.

>>11149346
Yeah, your suggestion led me to the correct answer. In this case the pattern seemed to be the gradual diminishing of the exponent to a minimum level and then back up again but with the exponent on y instead of x, with the first term being x^n-1 times y^0. It seems I had a gist of the correct idea at first but executed it poorly(for some reason I appear not to have considered there was a pattern to both x and y's exponents), it's one of those times I can't really explain why I didn't get it at first. Lack of attention, I hope.
Anyway, thanks again for the help.

BSc in Comp Sci + BA in Math
or
BA in Comp Sci + BSc in Math?

>>11149350
Listen, moron. This is not a thread for freshman courses and "weal anawysis wone". Go fuck off back to /sqt/ where you probably whine about trivial circuit diagrams and ask them.

>>11148639
>not an argument
you're right, i wasn't arguing anything. i was simply stating the fact that you have no idea what you're talking about.
would you like me to make a proper argument? okay.
topics such as haar measure on locally compact groups, general baire category theory, the space of continuous functions on compact hausdorff spaces and their duals (spaces of measures), maximal ideal spaces of function algebras, operator topologies and ideals in operator algebras via traces, and most CERTAINLY convexity in topological vector spaces (in full generality) and weak topologies (nonmetrizable) on banach spaces are all elementary topics that one covers in an introductory functional analysis course. if you do not fully understand, in generality, even ONE of these objects, your functional analysis course failed you tremendously.
perhaps you took a class intended for physicists? certainly, you can recite that the fourier transform is a unitary operator on L^2, but absolutely in the world of functional analysis, PDEs, operator algebras, etc. etc., give a single shit that you can do so.

>>11149554
>are all elementary topics that one covers in an introductory functional analysis course
Nope, one doesn't. It's most definitely stuff for advanced lectures, not something you would cover in the 4th semester.

>but absolutely in the world of functional analysis, PDEs, operator algebras, etc. etc., give a single shit that you can do so.
Did I say so?

>>11149538
Youre an incredibly sad fuck.
Unironically kill yourself, virgin.

>>11149479
PhD in Comp Sci + PhD in Math

>>11149577
Some of it is, but you absolutely should know about the continuous functions on compact Hausdorff spaces, Baire category and operator topologies.
You can't really call a Fun Anal course that's Hahn Banach, Arezla-Ascoli and Riesz a Fun Anal course.

>>11149585
why are you so touchy? are you insecure about something?

>>11149623
I certainly did some Baire in my fourth semester course, although if you want to talk about more topological results it would be appropriate to take a general topology course first, but anyway I was half meming anyway as I thought was obvious.

>>11147359
The set of countable admissible ordinals, that is the countable ordinals [math]\alpha[/math] such that [math]L_\alpha \vDash KP[/math], which is Kripke-Platek set theory, are unbounded in [math]\omega_1[/math]. The next admissible ordinal after the Church-Kleene ordinal is [math]\omega_1^{\texttt{ck}}(\mathcal{O})[/math], where [math]\mathcal{O}[/math] is Kleene's ordinal notation system for computable ordinals. In fact for any real [math]x[/math] you can define the set of ordinals which are the order type of an [math]x[/math]-computable well-ordering of the natural numbers. This ordinal will not be computable in [math]x[/math] and [math]L[/math] of that ordinal will model [math]KP[/math]. There is also the whole crazy world of recursively inaccessible large countable ordinals.

>>11149332
do not provide EE and CS brainlets with knowledge. just ignore these dumb insufferable fags.

>>11149645
>functional analysis doesn't require topology in his university
I'm guessing it also doesn't require measure theory.
Explains the fourth semester thing.

>>11149624
yes now fuck off

>>11149670
>I'm guessing it also doesn't require measure theory.
The basics of measure theory were taught in the third semester, in the stochastics and real analysis (Definition of the Lebesgue measure, L^p spaces and some manifold stuff, basically).

>Explains the fourth semester thing.
What is your issue exactly? There is a lot of functional analysis you can do with just L^p spaces, especially if you want to focus more on the direction of applications to PDEs which require more structure than just a topology, while obviously some measure theory is necessary if you want to go into deeper non-classical stuff.
That a fourth semester course doesn't cover clearly advanced topic should be unsurprising and that these topics are covered in advanced classes should also be entirely unsurprising.

>>11149288
None of these are challenging topics. If you understand the delta-epsilon definition of the limit and convergence, that is enough background.

Don't worry about the people who piss at you for being an engineer. They're wasting their energy seething while you try to better yourself.

>>11143961
Yes, there is no specification as to whether a and b are positive or negative here, and that’s what makes the proof so much simpler. To prove the triangle inequality you obviously need an inequality somewhere in the proof; the reason the second step is justified is very simple: 2ab <= 2abs(a)abs(b). Again, this is true regardless of the choice of a and b.

In the future please take your questions to the stupid questions thread, because frankly, these are stupid questions. And even if they weren’t, you should still post them there since this is basic material, and you’ll find more people willing to help you.

Very dumb question, I just proved that the span of a list of vectors is a subspace of the vector space that the vectors originally came from, but the next part of the proof asks me to show that it's the smallest subspace containing those vectors. What exactly do they mean by 'smallest'? My brain is screaming "take the case of R^2, but we can construct a subspace by taking the span((1,0)) where (1,0) in R^2, but span((1,0)) should have the same cardinality as R^2" but this is a common fact so I'm the retard here. Can someone set me straight here?

>>11149756
Take any subspace of the ambient space which contains those vectors. Because it's a vector space, it contains the span of all of its elements; in particular.

>>11149720
>there's a lot of functional analysis you can do with L^p
There is, yes.
>what is your issue exactly
There's a proper order to do things. Learning functional analysis ahead of measure theory and topology is just unreasonable. It's better to just put functional analysis in the sixth semester like a normal university.
>>11149756
>very dumb question
>posts it in /mg/ instead of in the very dumb questions thread
You lads are absolutely unbelievable.

>>11149766
Cut off the post accidentally. In particular, it contains the span of the list in question. So this shows that any vector space containing your list contains its span.

To prove S is the smallest set containing T, show any other set containing T contains S. That's what we mean by smallest.

Oh god I made my first comment on math stackexchange correcting some almost wrong answer and the author is angry because I pointed out a tiny error. Fuck karma greedy forums, it's much more pleasant to talk in anonymous anime board.

>>11149756
"Smallest" means two things in set theory: cardinality or inclusion. In this question they means "smallest" for inclusion. Here your must find that every subspace of R^2 containing (1,0) also contains span((1,0)).

>>11149756
Basically you want to say that, due to closure under scalar multiplication and vector addition, any subspace containing those vectors must contain their span.

>>11149769
Fuck off graduate student, this is /undergradmg/ now.

>>11149771
>To prove S is the smallest set containing T, show any other set containing T contains S. That's what we mean by smallest.
okay, I can work with this. Thank you for taking the time to point me in the right direction.

Oh god I made my first comment on math stackexchange correcting some almost correct answer and the author is angry because I pointed out a tiny error. Fuck karma greedy forums, it's much more pleasant to talk in anonymous anime board.

>>11149756
"Smallest" means two things in set theory: cardinality or inclusion. In this question they means "smallest" for inclusion. Here your must find that every subspace of R^2 containing (1,0) also contains span((1,0)).

>>11149769
>is just unreasonable
I think it depends what you want to do, for my 2 PDE courses all the functional analysis needed was covered prior, but now that I am doing a calculus of variation course the more abstract topological perspective seems to something far more necessary.

>>11149750
Thanks for an actually constructive answer.
Im not asking about the difficulty of the subject, im asking about how interesting/enjoyable the experience could be, what skills will I adquire, how differently its going to teach me to think, etc.
Im a senior year EE undergrad, at this point im pretty much fully trained to think like an engineer and that's wonderful, but i wanna go beyond that.
Ive had multiple math proffesors tell me to switch to math/take math classes throughout my uni years and im going to be able to finally go down that path next semester, so im excited like a little kid about it and want to get some insight from math veterans, thats why I ask here.

>>11149665
this hurts my head

>>11149665
>recursively inaccessible large countable ordinals.
excuse me?

>>11149614
I want to graduate before I'm 50

>>11149801
it's easier if you dont understand anything they said lol

>>11149669
They won’t do anything with it, you don’t change being an uncreative retard by studying sophomore math with watered down coursework and piss easy exams. Being defensive about sharing knowledge that’s freely available to the public is a sign of insecurity and a lack of progress on the part of that whole field and those studying it. If math is a living body of knowledge then idiots will never sully it by scraping at its necrotic outer shell for pseud points.

Anyone here teach math?
If so, what do you teach?
Is it enjoyable?
What do you think of your students?

are there any good books on proof theory for the uninitiated?

BSc Computer Science + BA Math. Y/N?

Light blue is Computer Science
Dark blue is Maths.
Green and Bronze are required Gen Ed courses.

>>11150309
>are there any good books on proof theory for the uninitiated?
Not maths.

I teach math. Primary College Algebra.
Not really enjoyable, I'd prefer to teach calculus 1.

90% of the students are shit because they are forced to take the course as part of their degree plan. Each year I hear droves of students go 'I'm just not a math person' and 'I'm not a genius like you' without knowing that I got to my level in mathematics by hard studious work and that none of it came naturally to me, so in reality what they think is a hybrid between a complement/please take pity on me phrase is taken as a deep insult so I want to slam their head against the cinderblock wall but don't because I'm much more afraid of the paperwork involved from administration then
the police inevitably showing up to take me into custody.

The remaining 10% are fine and I enjoy one on one talks with them. What I really like are the students who perform so well, that when they get a problem wrong, I worry that in truth I got the problem wrong so I have to double check everything.

>>11150297

I teach math. Primary College Algebra.
Not really enjoyable, I'd prefer to teach calculus 1.

90% of the students are shit because they are forced to take the course as part of their degree plan. Each year I hear droves of students go 'I'm just not a math person' and 'I'm not a genius like you' without knowing that I got to my level in mathematics by hard studious work and that none of it came naturally to me, so in reality what they think is a hybrid between a complement/please take pity on me phrase is taken as a deep insult so I want to slam their head against the cinderblock wall but don't because I'm much more afraid of the paperwork involved from administration then
the police inevitably showing up to take me into custody.

The remaining 10% are fine and I enjoy one on one talks with them. What I really like are the students who perform so well, that when they get a problem wrong, I worry that in truth I got the problem wrong so I have to double check everything.

>>11149585
i'm really not, i'm quite happy.

>>11150839
I believe him

>>11149577
>Nope, one doesn't. It's most definitely stuff for advanced lectures, not something you would cover in the 4th semester.
oh dearie me.
>>11149645
nah, it's not a meme when you out yourself as a complete filtered idiot early on. playing it off like "i was le pretending to be dumb" is kind of pathetic. you probably don't belong in analysis after all.
>>11149720
>PDEs require more structure than just a topology
certainly, but not only normed vector spaces. come on, jesus christ.
and stop calling these topics "advanced." they're not "advanced."
>>11149769
i appreciate your input, but none of the topics i mentioned were even in the least bit advanced, and basic point set + measure theoretic real analysis is at best a 3rd semester topic.
oh yeah, i always forget there are people who take fucking computational calculus in college.

>>11149798
fine. real analysis is fun. you'll get filtered by it and whine about your professor being "bad" and the book being "unclear." it's trivial to anyone with a future, like a simple child's game.
you're acting like our answers aren't constructive - i don't understand, all we're doing is telling you the honest to god truth. would you prefer us to lie?

>>11150488
good lord, what a pathetic schedule.

>>11150844
well, i sure don't.

>>11150853
>computational calculus
When people ask me what my beef is with engineers/scientists come from, it's the fact that they're the reason we have to teach computational calculus. This leads to a more then trivial amount of math majors who think that higher math is just fancier computational calculus while never having had to prove anything during their entire time in their freshman/sophomore years.

>>11150869
precisely. people who need to take any sort of math class which doesn't begin with "here are our initial definitions, here is our proposition, let's get to the proof" should be locked out of the mathematics major.

>>11150875
I don't hate those math majors, I just feel really bad for them. Imagine discovering a love for solving problems mechanically, so you decide to make that the focus of your career, only to discover that the last 2 years of mathematics were effectively a waste of time because what you have been doing is now relegated to computers who, unlike you, will always get it right, and now you're being asked to prove that there does not exist an integer that is both even and odd and you have no fucking clue what to do.

I'd argue that we should have Analytical Calculus courses geared towards Math Majors, but then the fucking engineers/scientists would invade and complain that the course is too hard

>>11150895
**Analytical Calculus courses geared towards Freshman Math Majors

>>11150136
A countable ordinal is called recursively inaccessible if it is admissible and a limit of admissibles. You can sort of port down large cardinal properties to large countable ordinals., there are notions of inaccessibility, Mahlo, compactness, among others. It is really interesting, Barwise's Admissible Sets and Structures is the best place to start.
>>11150148
He, don't confuse based higher recursion theory with category theory trannies.
>>11150309
Takeuti or Schutte, the handbook of proof theory is nice as well.

>>11149798
Real analysis consists of using the triangle inequality, the supremum principle, and delta-epsilon definitions. The undergrad class will not be terribly enlightening. I don't recall anything terribly technical in that class, though understand that it is many years behind me.

You would learn more from an advanced class on differential equations an advanced linear algebra class. This has abstract algebra as a prereq, but AA is incredibly easy; most of it is unpacking and repackaging definitions.

>>11150853
>filtered idiot
The only thing which "filtered" me was a numerics course.

>and stop calling these topics "advanced." they're not "advanced."
Yes, certainly only first semester stuff, what shithole u I do you go to where you don't do abstract topology in the first semester?

>>11150869
>while never having had to prove anything during their entire time in their freshman/sophomore year
LMAO. This is just an American thing, right?
Here any math major takes entirely proof based linear algebra and real analysis classes in the first semester...

>>11151294
In America, the first couple years are high school math.

>>11150297
>Anyone here teach math?
Yes, tutorials
>If so, what do you teach?
Bilinear algebra, euclidean spaces and Hilbert spaces
>Is it enjoyable?
Very much so
>What do you think of your students?
Good kids, well meaning but not very efficient. They have too busy a schedule (imposed on them by the uni) and most wind up not having time to do anything in depth.

>>11150853
>basic point set + measure theoretic real analysis
No, I'm talking about a proper one semester course on general topology, going through metric spaces, basic theorems, basic homotopy theory (i.e. up to Fraudenthal) topological manifolds and the classification of surfaces, and a measure theory course going through measure theory and probability.
Then again, you can just drag topology kicking and screaming into the second semester. First if the students are actually good, but it's tradition to wait for analysis.

>>11150859
Why are you so petty and why do you project so much? I guarantee ill do better at all the math classes ill take than you ever did at anything in your life.

>>11151211
I will end up taking those classes eventually, im just starting with analysis because the program is structured that way.

>>11151273
you're responding under the ridiculous impression that i'm lying to you or something. this is fucking sad, dude. i'm legitimately sorry that you're under the impression that these topics are in any way advanced or difficult.

>>11151294
obviously in america any math major with even the slightest chance at a future does the same as you. but people can choose to take a calculus course or two beforehand in their first year.
anyone who doesn't do proofs in an american uni in their second year is actually going to a shithole though.
>>11151342
that's just explicitly wrong. again, community "college" is not relevant here.

>>11151597
everything that you mentioned i did in my semester topology and measure theory course in my 3rd semester, besides topological manifolds (i did that in a separate differentiable manifolds course in the second semester) and probability measures (irrelevant).

>>11151810
i'm not petty or projecting at all. no need to make up these narratives in your head. again, merely stating facts which are based on heaps of empirical and inductive evidence.
here's what happened in my undergraduate elementary real analysis class:
- the course was all braindead easy for me and for a few other people in the class who knew what a fucking proof was
- the course was a nightmare for all of the engineering people who thought it would be fun to take le epic fancy math class
- many of them would get 30-40% on exams while i would get around a 94-97%
- they would constantly whine about how the lectures were unclear and disorganized and how the professor was intimidating
- the lectures were totally fine and honestly above average
i'm sure some of those kids still don't get the open cover definition of compactness, lmao.
and here's the kicker: my experience is the same as what i hear from EVERYBODY. every math person i talk to has the same experience in their real analysis class. sure, i bet some of them find it hard, but they don't make it this far into a phd if they do.
so again, not projecting!

>>11152049
I can tell from this post that you're a brainlet freshman that wants to feel superior to others.
All the freshmen that end up failing out of EE talk exactly like this about other engineers taking EE classes.