/sci/ - Science & Math

See how I linked this thread from the previous? Wasn't too hard now was it?

please no discussion of LEM, it been apriori proven thats the only way for the mind to work. Period.

>>11064763
That might be true, but even then it's not an argument to always adopt it. Proves that don't use it have an interpretation as programs in type theory, for example.

Does anybody have a good first principles reference on analytic continuation?
Or at least a collection of all magic tricks on how to find the continuations of common functions.
Seems, like combinatorics, to be a "bag of tricks" where having found one barely helps you next time.

>>11064771
>but even then it's not an argument to always adopt it. Proves that don't use it have an interpretation as programs in type theory, for example.
It just a way of weakening formal systems, is has nothing to do with the actual ontological status of LEM, you miss the point.

Did everyone miss this thread's edition?

What's /mg/ sexuality?
https://www.strawpoll.me/18801642

>>11064788
some people worship LEM like a god its quite cringe, they dont realize they are as dumb as trump supporting theists

To the TQFT waifu guy, can you post the (or a good one) reference to the stuff you're studying - I might reach into it before going to sleep. Something that's nicely written.

Logic is to most mathematicians what math itself is to physicists
A tool you take for granted and don't really want to see the complexities of, because you personally don't find them interesting

>>11065046
>physicists don't find math interesting

https://youtu.be/thnHe43S0z4
>tfw no IAS gf

>>11065052
meant that as an exaggeration
most physicists will not really concern themselves with the math they use

>>11064788
I would quite like it if each math general thread was numbered sequentially. If we call this thread #1 and the last thread #0, how far back do we go?

>>11065090
i've been in /sci/ for a while and if i recall correctly /mg/ as it stands now started around summer 2017, threads take 3 days to 10 days or so, so prolly in the small hundreds now

>>11065096
>3 days to 10 days
>last thread was 2 days
You lads need to post less.
>>11064805
>read it before going to sleep
Penrose.
Not that guy, btw.

>>11065112
ye tbf threads usually last a week but last few have lasted so little with all the logic shitposting

>>11065115
>shitposting
It's shitposting when it's math related and mildly decent, that's spam.

>>11065112
>Penrose
Does Penrose have something recent?

I mean the TQFT guy, he saw what he's currently reading. I looked up the last thread and it's about a Lieb-Schultz-Mattis theorem, which is somewhat messy. More messy than the standard topological Lagrangians with all dem A's and A^2's.
I thought I'd casually read up on theories without metric to get some more longer conversations going and I think he likes to talk (about his stuff).
I like LEM controversies like the next guy, but it's been degenerate in the last two threads.

>>11065123
lad im not entirely sure what ure trying to say

>>11064752
>no logic discussion allowed edition
Only because you said so?

>>11065159
It has nothing to do with math and invites pseud undergraduates, csniggers, and schizos.

Asking again, any textbooks recommendations about the geometry/topology of 4 manifolds? I have the one from Donaldson but I think I am still to much of a brainlet for that one

>>11065159
because if you havent noticed, the past few threads have been utter trash shitposts

>>11065178
t: Butthurt brainlet

>>11065185
They were great, unlike the dull previous ones. I, in particular, learned something new and saved those threads

Path integral / infinite integrals books?

>>11065190
Logic discussions between undergrads that first heard of LEM or constructive mathematics a week ago are not very interesting.

>>11065190
faggot

How do I get started with type theory? Can someone recommend me an introductory text and perhaps some papers I can survey?

>>11065217
https://github.com/jozefg/learn-tt
First textbook.
Avoid get consume by category theorist on twitter.

>>11065131
>https://arxiv.org/pdf/1907.08204.pdf
Actually, can someone recommend a nicely written text on the one-dimensional (spin-latice kind of) theories. I saw the transfer matrix solutions before, but I'm not set to only look at analytical problems either.
This is actually related to the question on analytic continuations, I want to get a feel for the sums and product involved in those issues.

>>11065190
How did you save them? What did you learn?

>>11065217
https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf

Discussions about LEM can be interesting, but not on /sci/ because it's the kind of topic that any idiot with zero mathematical knowledge can share their opinion on.
There's no barrier to entry that keeps out pseuds.

Fuck all this dumb logic and category talk.
*smacks book down on desk, flips to random page*
Let [math]\Delta[/math] be the maximal ideal space of a commutative Banach algebra [math]A[/math]. Call a closed set [math]\beta\subset \Delta[/math] an [math]A-boundary[/math] if the maximum of [math]|\hat{x}|[/math] on [math]\Delta[/math] equals its maximum on [math]\beta[/math], for every [math]x\in A[/math]. Prove that the intersection [math]\partial_{A}[/math] of all [math]A[/math]-boundaries is an [math]A[/math]-boundary.
You have 20 minutes.

>>11065217
>>11065225
I think the second (http://gen.lib.rus.ec/book/index.php?md5=015A853CBA3F01407E88B7B57B6F05D8)) would be better for someone with a math background interested in the more abstract, mathy side of tt.

>>11065369
What's the hat?
Does this require us to construct the ideal that the intersection is the boundary of?

>>11065404
Gelfand transform of [math]x[/math]

>>11065404
Hint: First show that there is an [math]A[/math]-boundary, [math]\beta_{0}[/math] which is minimal in the sense that no proper subset of [math]\beta_{0}[/math] is an [math]A[/math]-boundary.

>>11064788

If the OP did not want people to discuss logic, then he should not have posted an interesting picture with books related to logic. People are much more drawn to the picture of an OP, than its text. And when an OP chooses an image at odds with his text, he undermines the intent of his own thread, perhaps subconsciously. Typically, the image controls the conversation.

The text is what the OP "says". The image, what he "DOES". And actions speak louder than words.

*a-HEEEEMMMMM*
Let [math]P[/math] be a point in the interior of an ellipsoid of the Euclidean affine space [math]\mathbb{R}^{3}[/math] Draw three orthogonal straight lines through [math]P[/math], and let [math]A_{1}, A_{2}[/math] be the intersection points of the first line with the ellipsoid; [math]B_{1},B_{2}[/math] the intersection points of the second line with the ellipsoid; and [math]C_{1},C_{2}[/math] the intersection points of the third line with the ellipsoid. Prove that the sum:
[eqn]
\frac{1}{PA_{1}\cdot PA_{2}} + \frac{1}{PB_{1}\cdot PB_{2}} + \frac{1}{PA_{1}\cdot PA_{2}}[/eqn]
does not depend on the three orthogonal straight lines chosen.

>>11065430
shut the fuck up nigger

>>11065438

I have written a very reasonable and true thing, and you know it. If you had a better rebuttal, you probably would have made it.

Aaaannnd one more
*sits back in chair preparing to light another fine cigar*
Let [math]a_{k,n}[/math] denote the minimum number of coefficients in the pattern inventory of all [math]k[/math]-colorings of the corners of an [math]n[/math]-gon needed so that using symmetry all other coefficients in the pattern inventory are known. Find a generating function [math]g_{k}(x)[/math] for [math]a_{k,n}[/math].
chop chop

>>11065437
You fucked the Tex up.
Anyhow, I get the impression that the problem is showing that SO(3) preserves that value because it's either the ellipsoid's volume or some multiple of it.
I'm not computing that tho, miss me with that shit.

>>11065460
looks fine on my end

>>11064805
>>11065131
Hello, the TQFT Yukariposter here. Penrose has little to do with much of recent development in TQFT; CFT through string theory perhaps, but not nearly to the extent where he would have a reference.
In general, LSM/HOLSM-type (Hastings-Oshikawa) theorems give necessary conditions for the existence of gapped symmetric ground states (GSs) in generic fermionic systems (liquid or otherwise), and is in fact a very powerful tool for understanding how topological phases arise. The short-range entangled and tensor network states responsible for topological phases in (1+1)D are examples of such symmetric gapped GSs.
The reason why HOLSMs are much more powerful than Chern-Weil (when applied to Chern-Simons-like, [math]F[/math] and [math]F^2[/math], theories) is because it is non-perturbative, in the sense that the topological properties of the GS can be extracted directly from [math]ab initio[/math] interacting models instead through the (dubious) construction of an effective topological field theory Lagrangian first. Besides, non-liquid topological phases and those with sub-extensive GS degeneracy in (2+1)D have no well-defined continuum limit, and hence no notion of a "TQFT". HOLSMs can tackle these cases.
Of course, this comes with the price of not knowing exactly what the topological invariant characterizing such phases are, but works toward anomalous textures have proposed twisted equivariant/group cohomology theories as candidates. This, as well as understanding the bulk-edge Gysin map on them, is my current project.
>>11065227
For foundation, I have found Lehmann's "Mathematical Methods of Many-Body Quantum Field Theory" and Sénéchal's "Theoretical Methods for Strongly Correlated Electrons" (first few chapters) to be good reads, especially for those with a mathematical slant.

>>11065463
You repeated 1/(PA1 \cdot PA2)

>>11065468
oh shit so it is. it's how it's written exactly in this textbook too.

Bonus round: god mode.

An [math]R[/math]-module [math]M[/math] is 'locally free' if [math]M_{p}[/math] is free as an [math]R_{p}[/math]-module for every prime ideal [math]p[/math] of [math]R[/math]. Prove that a finitely generated module over a Noetherian ring is locally free if and only if it is projective if and only if it is flat.

>>11065466
Some advances in quantum optics, photon matter interaction, something like scattering photon in QFT.

oh, I should leave a problem that janny can do so he doesn't feel left out:

Find the roots of [math]p(x)=x^{2}-4x+3[/math].

>>11065527
Why should I bother finding them if I know they exist (in C)?

>>11065564
because I asked you for them

How do physicists even tank this autism?
>>11065564
Why even bother constructing theorems when they exist in a topos?
>>11065527
>integer solutions
Very nice of you, I'm sure janny will appreciate it.

How are they simplifying this? I don't understand.

>>11065586
try reviewing your exponent rules and then approach the problem again

>>11065225
Based post

>>11065589
I figured it, thanks though

>>11065444
>le debate me on whether I’m a needy faggot
You seem lost

>>11065571
Legendre transform, as well as many operations that physicists do that seem "questionable" to the untrained eye, is made rigorous by contact geometry. As long as [math]T^*M[/math] forms a symplectic manifold, the contact manifold can be constructed from pulling back a line bundle [math]L \times T^*M [/math] by some smooth map [math]f:M\rightarrow M[/math] and everything follows.

>>11065639
Legendre transform doesn't seem "questionable" at all

>>11065639
Hit me up with an appropriate text, please.
>>11065652
No, there are some extremely serious problems with functions not having domains in that picture.
Extremely serious and confusing.

>>11065652
Sure it does, especially in the QFT context where you Legendre transform the interaction vertex to get the quantum action. This requires a symplectic (or even Poisson) structure on the infinite-dimensional jet bundle.

>jet bundle
is it as cool as it sounds?

>>11065667
No.

>>11064191
the computables form a countable set (inject into set of finite strings with a program that computes each). countable sets are meager, i.e. contained in a countable union of closed sets with empty interior (merely take each point on its own). the complement of a countable union of closed empty interior sets is a countable intersection of open dense sets, which by baire category is dense.

>>11064797
>all these disgusting str*ights
all destined to be actuaries or software "engineers." incapable of ever making it.

>>11065420
that's not a hint, that's literally the fucking problem statement you moron.

>>11065661
https://arxiv.org/abs/1604.08266
Incidentally, symplectic geometry underpins Lagrangian mechanics. For an application of contact geometry to thermodynamics, you can also look up "geometrotheromodynamics" by Quevedo.
>>11065667
I don't know, is the bundle of fields cool to you anon?
>>11065682
Lol.

>An urn contains 23 white beans and 34 black beans. An African man takes out two beans; if they are the same, he puts a black bean into the urn, and if they are different, he puts in a white bean from a large heap he has next to him. The African man repeats this procedure until there is only one bean left. What colour is it?

>>11065690
Don't be mad at me that you can't figure it out

>>11065692
>I don't know, is the bundle of fields cool to you anon?
Give me the undergrad version rundown.

>>11065369
Have you tried induction on that weird B symbol

>>11065704
We take a [math]k[/math]-jet to be an equivalence class of functions [math]\varphi,\psi \in C^\infty(M)[/math] on a smooth manifold [math]M[/math] such that [math]\partial^r \varphi = \partial^r \psi[/math] for [math]r < k[/math] (up to a constant factor). The point-wise equivalence relation defines a vector space of [math]k[/math]-jets, and this forms the fibres of the [math]k[/math]-jet bundle [math]J^kM[/math] on [math]M[/math].
We then take the inductive limit [math]k\rightarrow \infty[/math] to get the jet bundle [math]J^\infty M[/math]. Intuitively for analytic functions, fibres are basically "evaluations of fields and their derivatives".

>>11065729
Yeah, that's pretty neat I guess.

>>11065746
It makes formalizing Lagrangian density straightforward. [math]\mathcal{L}[/math] is just a [math]d[/math]-form on the jet bundle, where [math]d = \operatorname{dim}M[/math]. The action is then "just" the "de-Rham" pairing [math]\langle M,\mathcal{L}\rangle = \int_M \mathcal{L}[/math].

>>11065178
>>11065190
>>11065190
>t. butthurt brainlet

He's right though. Discussions about logic inevitably attract pop sci pseuds who want to talk about Godel, philosophers that often aren't actually interested in math, and undergrads who just discovered that logic consists of more then just the rhetorical strategies their English teacher taught them about in high school.

Also, there isn't too much logic research going on today IN MATH DEPARTMENTS. There's actually been a huge resurgence of interest in proof theory, modal logic, and algebraic logic in the last 15 years or so, but it's mostly gets attention in CS, economics, and cognitive science departments.

>>11065704
this has already be answered, but I suggest to actually learn what is going on you check out Olver's "Applications of Lie Groups to Differential Equations", which is in my opinion the best introduction to working with jet bundles and general Lagrangian mechanics, in particular the proper versions of the Noether theorems

>>11065794
I'll check it out, thanks.

Currently taking algebra, which is easy as fuck and I'm currently 6 weeks ahead of the course in reading/homework.

I want to retake the placement exam (sober this time) and skip trig/precalc. What book should I use to teach myself those two subjects in 2 months?

>>11064752
Can the principia be used to ensure your fundamentals are rock solid or should it only be tackled after you have some undergrad math under your belt?

>>11065700
this nigga picking beans

>>11065855

If you think you can handle it, Serge Lang's Basic Mathematics is a decent book for someone looking to have a solid grasp of high school mathematics. Not that great for teaching yourself the super hard questions and techniques to solve shit.

Another good choice is Paul's Online Notes: http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx

This will teach you all the pre-calc shit, but no trig. You'll have to find some other resource for trig (I am currently looking for one myself).

>>11065692
>https://arxiv.org/abs/1604.08266
Nice shit.
Extremely intuitive, too. I get the impression the entire thing was built up so that the Reeb vector field pointed in the direction of time and the restriction to some fixed time gave the usual symplectic hamiltonian.

>>11065894
The high school math I think I'll be fine in. I'm almost done with the Algebra book and just aced my midterm. It's mostly been a refresher for me.
>You'll have to find some other resource for trig (I am currently looking for one myself).
My college recommends this:
https://www.pearson.com/us/higher-education/product/Dugopolski-Trigonometry-Loose-Leaf-Edition-Plus-My-Lab-Math-with-Pearson-e-Text-24-Month-Access-Card-Package-5th-Edition/9780135268575.html?tab=contents

Without access to homework/test problems to practice on, I dunno how good it will be, though.

>>11064771
>programs in type theory, for example.
>>>/g/. Subhumans aren't welcome in this space.

>>11065633

>my trips >> your dubs. My truth value is greater than yours. Again, if you had anything apart from ad homs, you likely would have deployed them. I win.

>>11065701
no, i can't figure it out yet, and i'm mad cause i'm taking banach algebras right now

So I'm trying to build my own self-study regimen. I'm curious about a bunch of different topics on a surface level, but for a more long term goal I'm looking towards mathematical physics.
Obviously the major topic here is differential geometry/topology right? I'm studying John Lee's Introduction to Smooth Manifolds which seems quite comprehensive.
How important is functional analysis? I'm currently learning about measure and integration from Folland's real analysis text, and I look forward to the chapters on functional analysis (it seems like a pretty cool topic actually).
What about algebraic topology? I'm not interested in going super deep into that stuff, but I want to learn enough to understand the basics of homology and cohomology, and understand some of the more digestible results. I'm learning about some of the basic homotopy/fundamental group results, and its all pretty cool (I love seeing fancy applications of algebra).
Of course, there's other stuff I want to learn, like Galois theory (so cool), and complex analysis (specifically for some of the more famous applications to number theory). I'm largely staying clear of studying algebra purely as an end in itself (it has to be motivated by something like topology for me to really care, exception being Galois theory I guess). Differential equations don't thrill me that much either, but it is cool when they motivate interesting topics like Fourier analysis.

>>11066202
functional analysis and differential geometry are very very important
functional analysis is the math behind quantum mechanics. differential geometry is the math behind relativity.
algebraic topology is not that important but it's useful for differential topology/geometry and anything with manifolds. i wouldn't prioritize it but it's fun.
galois theory sounds worthless to what you're doing, but you should do things which interest you.
i would recommend looking into proper PDE theory and dynamical systems theory after doing some measure theoretic real analysis and some functional analysis. ideas like distribution theory, ergodicity, and unbounded (differential) operators on hilbert spaces are really central to much of mathematical physics. DEs at this level are not "lol solve le equation," it's much more like normal analysis.

>>11066121
>three 4’s
>misuses greentext and thinks this is a debate
You need to go back friend

>>11066202
Diffgeo is extremely important, and Lee's book is nice. Functional analysis is also very important, I would suggest "Quantum Theory for Mathematicians" by B.C. Hall for a nice physics-oriented introduction. AlgTop & universal algebra stuff in general is NOT your primary concern when starting to learn the theory.
As a general rule of thumb, study what you enjoy most, can't state that often enough. You might think subject a sounds super nice but during your studies you realize that subject b is actually what you enjoy. Don't buy into any kind of elitism of the sort "applied math sucks lol I'm pure".

How is type theory different than set theory?
How is category theory different from abstract algebra?

>>11066202
Mathematical physics is almost Functional Analysis over PDE in Diff geometry.

Lie Algebra over PDE or Manifold.
For lie algebra and representation theory looks this intro than become in full book
Lie Groups, Lie Algebras, and Representations - An Elementary Introduction Brian C. Hall
https://arxiv.org/abs/math-ph/0005032

>>11066223
>How is type theory different than set theory?
I have no idea, really. Your best bet for an actual answer is studying both. Good luck
>How is category theory different from abstract algebra?
Category theory is a language. It's usefull in some parts of math like algebra. Algebra is a very broad mathematical field.

>>11066215

Still missing the point.

>>11065213
Ala physicists you should check fujiwara book. If you want proper infinite dimensional integration, stochastic process is the key.

>>11066242
Not him, but got a rec for a book on stochastic processes?

>>11066242
>If you want proper infinite dimensional integration, stochastic process is the key.

Like Stochastic / Integral Geometry?

Thanks.

>>11066249
integral geometry = integer valued geometry
careful

>>11066257
Supposedly this book
https://www.springer.com/gp/book/9783540788584

Move Radon transform to higher dimensions and using geometric measure theory.

>>11066262
careful

>>11066262
What's the difference between "stochastic" and "information" geometry?

>>11066266
ok
thanks

>>11066212
>>11066219
>>11066227
Thanks. This is really cool to hear. I'm now much more excited about learning Functional Analysis.
I am now excited about learning PDE's "for real" once I get more analysis under my belt, because the way that the topic is taught to undergrads is very uninteresting and has caused me to have a negative perception of the topic.

>>11066274
I'm an idiot, just read some stuff and now I get it. No need to answer my question.

How do I start studying inter-universal teichmuller theory?

>>11066291
If you have to ask you are not ready to study it. Unironically.

>>11066291
IUT is a literal cult. Next you'll be saying Atiyah solved the Riemann Hypothesis

>>11066303
he did though

>>11066281
PDEs "for real" is just functional analysis on fancy spaces called sobolev spaces. if you end up liking functional analysis then you'll probably like pde (though the flavor is very slightly different from more standard functional analysis)
another direction you can go with functional analysis is toward operator algebras, like banach algebras, C* algebras, and von Neumann algebras. all relevant to physics. but for now just looking at Lee for diff geo and whatever for FA is totally fine

>>11066303
>IUT is a literal cult.
Millions of sheep believe in various cults like Catholicism, etc. How is this any worse? Surely it can only be better because it's more rationalistic in nature.

What's a good thorough book on the history of mathematics?

>>11066332
Why don't you read it and find out?

>>11066336
good one

I'm stuck at an abstract algebra problem... Let t<n, show that there exists an injective homomorphism from Z_t to S_n.
So far I've trying with [k] to a k-cycle but it doesn't work, I've also tried with transposing the kth element but it doesn't work either

>>11066348
think about it this way. to embed Z_t, what length cycle do you need to find in S_n?
okay, now what's the first cycle you think of with that length?

>>11066332
Katz.

>>11065571
Legendre transformation is not really that interesting. Only useful property is that it keep concavity.
Which is fundamental fof most functions in statistical physics.

The intellectual effort of physics is less to prove the shit you do, and more to give it an actual physical meaning.

>>11066385
Got it, thanks!

>>11066223
Many instructions that are set comprehensions in set theory are (but not only) primitive in type theoretic (product, disjoint sum, function space) and those come with explicit construction rules. The language then has not only proposition but also judgements. E.g. you can make the judgement
(3,7) : N x N
where 3 is short for SSS0 and N ist defined (effectively axiomatically, if your theory is so weak you can't model it) as (rather generic) type with only two "axioms"
0 : N
S : N -> N
A statement of the form "a : A" (a judgement) is not a proposition you can use as subexpression in other sentence.
The type of a term is unique in this framework (unlike set theory, where both {3, N} And {{3,5},3} are set containing 3)
Type theory isn't a mathatical theory written down in logic. Instead, the theory is a logic. And more, it's a proof relevant logic.
And it embeds other logics . E.g. considering the type of Propositions Prop and consider two generic propositions (Think P:Prop and Q:Prop and R:Prop and universally quantify over them, i.e. consider P and Q variables in the following), then
((P->Q) x P) -> P
is inhabited (the term is "(f,p) mapsto ,f(p)")
And so is, for example)
(P->R x Q-> R) -> ((P + Q) -> R)
(via "(f,g) mapsto ((p or q) mapsto If (p typecheck against P) then return f(p) else return g(q)))")

The first example is modus ponems, the second says that "and" is stonger than "or".

All mathematical propositions have a type in that sense that's equivalent to them (at least with term axioms). All intuitiinistic propositions are constructible (have algorithms in terms of functions, pairs and sums for implications, and and ors) unconditionally (see Coq).
Google for Curry-Howard

can someone tell me what happens to the (x-2)^-1/2? I posted earlier and thought I had it, but I had the wrong stuff written down. It just seems to disappear. If the -1/2 means 1 over the square root of (x-2), where does it go? Wouldn't it need to become (x-2)^1/2 in the denominator and then combine to 3/2 with the other like base?

>>11066202
>How important is functional analysis?
It is the most important thing to know if you care about the non classical theory of PDEs.

>>11066316
>PDEs "for real" is just functional analysis on fancy spaces
Well, there is also a lot of classical theory on PDEs which obviously isn't accessible to FA methods.

the second part is easy, its just the mean value theorem, but the first part i cant puzzle out. i feel like using the intermediate value theorem/rolle's theorem is the way to go but the details elude me

>>11066595
not rolle's theorem i dont know what i was thinking there

>>11066438
Why exactly is the Axiom of Choice purely in [math]\text{Type}[/math] in Coq weaker than that in [math]\text{Prop}[/math]. The latter needs to be stated, whereas the former can be proven/constructed as a suitable type

I'm having trouble understanding how transvections generate the special linear group.
You need to do it by induction, but I don't really understand how they "separate" the hyperplane of (n-1) dimensions and a line.

>>11066438
>ist
Guten Abend, mein Freund.

>>11066595
I think I have a proof, but it is ugly.

>>11065481
Are we assuming commutative here?

>>11066291
>How do I start studying inter-universal teichmuller theory?
See section 3.2 of https://www.maths.nottingham.ac.uk/plp/pmzibf/notesoniut.pdf

>>11065438
>nigger
Why the racism?

Threadly reminder to work with physicists.

>>11065466
>>11065639
>>11065663
>>11065692
>>11065729
>>11065753
Why are the files deleted?

>>11066202
>So I'm trying to build my own self-study regimen. I'm curious about a bunch of different topics on a surface level, but for a more long term goal I'm looking towards mathematical physics.
>Obviously the major topic here is differential geometry/topology right? I'm studying John Lee's Introduction to Smooth Manifolds which seems quite comprehensive.
>How important is functional analysis? I'm currently learning about measure and integration from Folland's real analysis text, and I look forward to the chapters on functional analysis (it seems like a pretty cool topic actually).
>What about algebraic topology? I'm not interested in going super deep into that stuff, but I want to learn enough to understand the basics of homology and cohomology, and understand some of the more digestible results. I'm learning about some of the basic homotopy/fundamental group results, and its all pretty cool (I love seeing fancy applications of algebra).
>Of course, there's other stuff I want to learn, like Galois theory (so cool), and complex analysis (specifically for some of the more famous applications to number theory). I'm largely staying clear of studying algebra purely as an end in itself (it has to be motivated by something like topology for me to really care, exception being Galois theory I guess). Differential equations don't thrill me that much either, but it is cool when they motivate interesting topics like Fourier analysis.

>>11065090
>I would quite like it if each math general thread was numbered sequentially. If we call this thread #1 and the last thread #0, how far back do we go?
Sequentially numbering generals is reddit.

>>11066673
whats the outline?

Since mathematics is functionally the study of the category of [math]ZFC[/math]-theorems, aren't we all category theorists?

>>11065894
>If you think you can handle it, Serge Lang's Basic Mathematics is a decent book for someone looking to have a solid grasp of high school mathematics.
Lang is a meme.

>>11066710
Subtract both sides and view it as an equation in x0, obviously it is negative and positive for some values of x0 (if f isn't the zero function) intermediate value theorem, qed.

>>11066241
There is no point, its not a discussion. Nothing you’ve said was substantive and you’ve misinterpreted the nature of the interaction. Off-topic leddit tourist posting is against the rules, fuck off.

>>11066714
ZFC is a subset of mathematics, so is category theory.

>>11066764
>ZFC is a subset of mathematics, so is category theory.
Category theory is a supercategory of mathematics.

>>11066764
>ZFC is a subset of mathematics
>>>/lit/.

>>11066177
don't feel bad bro, it's from grandpa rudin

>>11066611
https://coq.inria.fr/library/Coq.Logic.ChoiceFacts.html
Maybe this helps you. "Choice" means a million different things.
I for one, think choice should hold in a theory of sets (that is, if you want a theory of collections in the way your granny conceives of collections), but not the power set axiom generically.

>>11066644
Mahlzeit

>>11066464
[math]x^{a}\cdot x^{-a} = x^{a+(-a)} = x^{0}=1[/math]

>>11066202
>PDE’s
>Functional Analysis
>Diff Geo
>Fourier Analysis
most of the god tier useful maths desu
>alg top
>galois theory
trash

Post less or post better, lads.
>>11066812
Alg top is extremely useful for diff geo.

What's the deal with category theorists on twitter?

https://www.codechef.com/IEMATIS1
>IEMATIS1 - International Mathematics Olympiad
>There will be 5 levels and each level will have one question.
>The answers will be integer type and each participant will have three chances to lock an answer for a particular question. After three unsuccessful attempts their id would be blocked and they cannot continue participation.
>Level of question will be of International Mathematics Olympiad(IMO) level.

>>11066595
Since [math]f[/math] is continuous on a compact interval, it is bounded above and below by two values it attains, say [math]f(c_0)\leq f(x)\leq f(c_1)[/math] with [math]c_i\in I=[a,b][/math]. Therefore since [math]g\geq0[/math], we have that [eqn]\int_I gf(c_o)\leq\int_I gf\leq\int_I gf(c_1)[/eqn], and in particular, this inequality works the same if we take [math]f[/math] outside the second integral since we can do the same with the two outer integrals (since [math]f(c_i)[/math] is just constant):
[eqn]f(c_o)\int_I g\leq f\int_I g\leq f(c_1)\int_I g[/eqn]

Now just use IVT

>>11066693
not a fan of the blacks

>>11066876
>76>>11066595
ignore this post, i wrote it when i had just woken up

>pick up Tu's diff geo book
>good explanations and presentation
>the exercises are in my sleep tier
God damn it.
By the way, I was looking thorugh Verbitsky's page and I found the following problem: if two manifolds have isomorphic C^infty rings, they're diffeomorphic.
I can't come up with a proof that isn't essentially being autistic with compact supports and showing that they have isomorphisms of locally ringed spaces by considering bump functions.

>>11066768
>The undergrad category tranny pseud is at it again
Just fucking kill youself already you mentally ill tranny. Nobody is impressed that you can parrot abstractions about thing you do not understand.

>>11066832
I only know of Its and Beaz.
You probably mean Scala trannies. When they say category theory, they mean commutative group theory and mapping over lists.

>>11066611
>>11066792
TL;DR of 4 posts: The axiom of choice as you know it is a way to relate the logical quantifiers with the local notion of function.
Its power depends on how strong your quantifiers are, how slack your definition of function is, and how they relate to each other in the first place.
Usually, the mismatch lies either in the existential quantifier hiding information, or the universal quantifier providing less information than needed to form a function.
When such a mismatch exists, AC postulates that you can throw a tantrum and pretend that the information exists.

The axiom of choice is a statement about families of subsets of a set.
Given a family of inhabited subsets of a set, there exists a function from the index set to the set such that for all indexes it picks an element of the appropriate subset.
Sets, subsets, families of sets and families of subsets of a set are all notions that are mostly conflated in material set theory, in which sets can have arbitrarily deep nesting, but in type theory you only get as much "nesting" as you ask for, so we pick:

Set = Type.
Subset of S : Set = A predicate on S, S -> Prop.
Families of sets = A function from the index set I, I -> Set.
Families of subsets of a set = A function assigning for each index from an index set I a subset of S, i.e. into a predicate on S, so I -> (S -> Prop), i.e. a binary predicate (I, S) -> Prop.

1/4

>>11067002
Given all of this, you're ready to read the axiom of choice in type theory as:

S is a set.
F is an I-indexed family of subsets of S.
S : Set
F : (I, S) -> Prop

Each subset is inhabited, thus for all indexes i, there exists an element s : S satisfying F(i,s).
[math]\forall i : I. \exists s : S. F(i,s)[/math]

Then, there exists a function f that picks for any index i, an element in S satisfying F(i,f(i)).
[math]\exists f : I \to S. \forall i : I. F(i,f(i))[/math]

If [math]\exists[/math] is [math]\Sigma[/math], the strong constructive existential quantifier that gives a pair of a witness and a proof that can be independently manipulated, then the statement is trivial:
The proof of inhabitation of each F(i,s) is a function from an index i, to a pair of s : S and a proof that it satisfies F(i,s), and you are asked for a pair of a functions, one a plain function f : I -> S, and the other a function from an index i to a proof that F(i,f(i)). In each function you just apply the original function and project out of the pair what you need, forgetting the other: one returns the witness, the other the proof. Since \Sigma gives a strong, global guarantee that the elements of the pair stay related even when separated, the two functions are automatically related.

2/4

>>11067004
If [math]\exists[/math] is a weaker existential quantifier, in which the proof and/or the witness cannot generally escape the scope of the elimination of the existential quantifier, or are required to be used together, then you can't split the function anymore.
Usually to restrict values within a "scope" you would use a monad, and indeed many, but not all, such existential quantifiers can be recovered as compositions of [math]\Sigma[/math] with a monad M, AC for M would be a statement about the commutation of M with the universal quantifier. If M = double negation, then you get the classical existential quantifier.

[math](\forall i : I . M (\Sigma s : S. F(i,s))) \to M (\Sigma f : I \to S . \forall i : I . F(i,f(i)))[/math]

Note that it didn't matter whether Prop = Set = Type, i.e. whether we were using propositions-as-types or had a different type for propositions.
If we take propositions as types that have at most one value, whose only content is whether they have a proof or not, then note that they are not closed under [math]\Sigma x:A. P(x)[/math], as it usually has many proofs even if P is a Prop-valued predicate, in which case it would be the type of elements of A satisfying P, like a subset type. So, strictly speaking, it is not a propositional existential quantifier.
If you take M = a propositional truncation, such as double negation, then you recover a propositional existential quantifier with "at most one proof", but lose AC, as the information cannot escape the scope of the quantifier anymore. If you have a Prop type in the first place, you'll likely want such an existential quantifier lying around, so this is probably the AC you were referring to.

3/4

>>11067009
There's one more way in which AC can fail, by mismatching the definition of function and universal quantifier like in material set theory.
Rather than taking sets as types and functions as type theoretic functions, you take sets as types equipped with an equivalence relation, and functions between sets as type theoretic functions that are extensional relative to their equalities.

Rather than having

Set = Type, Func(A,B) = A -> B

you have

Set = a pair (X, ~) such that X : Type and ~ an equivalence relation on X, noted x ~ y : X for x, y : X
Func(A, B) = f : A -> B such that x ~ y : A maps to f(x) ~ f(y) : B

Now AC asks for an extensional choice function, but you never redefined the universal quantifier to be extensional, so you're lacking the necessary information to produce the proof of extensionality and are fucked. If you restrict the universal quantifier to be extensional, you get the setoid model of extensional type theory.
This post is long enough already so read http://archive-pml.github.io/martin-lof/pdfs/One-hundred-years-of-Zermelo-s-axiom-of-choice-what-was-the-problem-with-it-2009.pdf for more information about it.

4/4

>>11066969
wym by exercises are in ur sleep tier, too hard or easy?

>>11066935
whats wrong with that proof
it looks right to me

Is it safe to use the Axiom of Choice if I care about coherence and rigour in maths? Sometimes when using it it feels like using false magic and handwaving away something you should give much more thought to. Does anyone else feel this way or do you just learn to deal with it as you get more experienced?

>>11067038
Ignore that reply, i wrote it when i had just woken up from a nap

>>11067047

>>11066542
Sure, there are much more dynamical ways to study PDE, and often these need to be brought together to get nontrivial results. Even when PDEs are studied classically though, I still think that doing functional analysis "feels like" doing PDE (which makes sense because both are quite literally linear and nonlinear algebra on hilbert/banach spaces, and you tend to just introduce more explicitly analytic objects in the classical theory of PDE but it really all comes together imo)

>>11066812
Algebraic topology isn't trash Geo/PDE bro, it's the most beautiful kind of algebra because it isn't about algebraic things at all. It's a slave to topology.

>>11066969
Early diffgeo exercises are always like that, don't worry. The later you get, the more interesting they should get.

>>11067045
>Is it safe to use the Axiom of Choice if I care about coherence and rigour in maths?
Yes, don’t fall for the memes
>Does anyone else feel this way or do you just learn to deal with it as you get more experienced?
Sometimes there is a way around it, sometimes there is not. You should worry about that after you have given and understood the proof with your full toolbox.
Otherwise it’s the classic "premature optimization" procrastination technique. In other words, learn to walk before you start worrying about doing handstands

>>11067020
I can solve it like I can solve arithmetic or evaluate integrals.
It doesn't even require any geometry.

>>11067045
>>11067136
axiom of choice is a path for brainlets, you use it because you can't provide the explicit construction and in the end it leads to paradoxes such as Banach-Tarski and/or well-ordering of sets. The only reason why people are still using it is because they don't want to be bothered with details.
Same goes about LEM ideally you can do math without (I mean not completely dump it but only use it for instances where you've proven it holds) but that is infinitely harder to do.

>axiom of choice is a path for brainlets,
enjoy your vector spaces without basis.

I think anon in this or last thread made a good point, I think we should start numbering these threads. There would be more motivation to link the previous thread and would keep a better order to track good posts.

I propose the next thread starts as /mg/ #300

Retard here

How do I solve an Mean Value theorem for an absolute value?

>>11067274
status.fireden.net records about 184 threads, but this only goes back to 2017. To make things easy why not just make the next thread number [math]\aleph_{0}[/math]?

>>11067275
What does that even mean?
Has it any relationship to your picture?

>>11067292
I need to solve it with a mvt for integrals. Khan Academy doesn't exercises for that specific case, only mvt for normal integrals.
It says the x value lies between minus to + infinity.

>>11067207
>paradoxes
>implying
banach tarski isn't even a paradox. of course if you break something into parts which don't have a sensible meaning of volume, you can do whatever you want with them. You can split a ball into a ton of points, and each has volume zero. Is it a paradox that the ball doesn't have volume 0? No, because the rules of measure theory ask for countability. But this is essentially the same as what you're doing for construction of nonmeasurable sets, doing uncountable operations on things which work countably. It's not strange that this causes problems, and it's only unintuitive until you know the most basic measure theory.
I would say the same is true for well ordering of sets. Sure, it's strange. But only before you see the ordinals properly defined. Then it's sort of pedestrian, and you realize the person you were talking to was trying to pull the wool over your eyes by pretending R had any significance in the statement "try to put a well ordering on R, see what happens." This automatically gets you thinking about "nice" orderings that are okay with the field structure or topology. Of course not.
The reason I can't write one down for you isn't because it's magical or unknowable. It's just because it's inherently uncountable in nature, and I can't write down things which require more than finite length strings to encode. Just like the uncomputable reals, just like tons of other things which I'm sure don't bother you.

>>11067298
>I need to solve it with a mvt for integrals. Khan Academy doesn't exercises for that specific case, only mvt for normal integrals.
The mvt applies to all continuous functions, not just the ones expressed by integrals.

>It says the x value lies between minus to + infinity.
If I didn't know I had guessed or asked, but it is neither a question not a claim, just a definition.

>>11067316
Oh, yes, sorry, dumb definition of mine. Of course the mvt is the mvt for all functions.

Anyway, am I correct that when it goes from plus to minus infinity, that 1/avg is 2 pi? At least it was like that in the signum function I did.

>>11067259
The axiom of choice matters for vector space bases exactly in the cases where you can't state a basis. That's exactly one of the most persuasive examples again the axiom of choice.
>vector space for which humans can probably not state what a basis of it would be
>by the axiom of choice, there exists a basis. Let's prove some other existence theorems based on that, we'll use end up with nice theorems about all the other nice things with "exists"

>>11067259
>why yes, I study infinite-dimensional vector spaces without topologies or further structure
I too wish there were devil-like people studying such an absurd monstrosity of a field, but there aren't.

>>11067259 #
The axiom of choice matters for vector space bases exactly in the cases where you can't state a basis. That's exactly one of the most persuasive examples against the axiom of choice.

>vector space for which humans can provably not state what a basis of it would be
>by the axiom of choice, there exists a basis. Let's prove some other existence theorems based on that, we'll use end up with nice theorems about all the other nice things with "exists"

>>11067359
Axiom of choice is strictly equivalent to Zorn Lemma/Hausdorff Maximality theorem, which is ubiquitous in functional analysis and measure theory.
The results may not be "positive", (there exists an infinity of non-measurable subsets in any set of nonzero Lesbegue measure, iirc), but they are interesting in their own way.

>>11067321
>1/avg is 2 pi
What is "avg"?

>>11067355
Can you prove that any normed Vector space has a basis without AC?

>>11067376
Yeah, I don't really have a problem with choice. It's natural for set. (Although I'm not a big fan of it in strong theories such as Z or ZF.)

>>11067392
1/f(b)-f(a), since b is + infinity and a is minus infinity, I assume 1/avg (avg for average) is 2 pi.

>>11067410
No, but you don't want a basis when you're working over a normed vector space.

>>11067413
>1/f(b)-f(a), since b is + infinity and a is minus infinity, I assume 1/avg (avg for average) is 2 pi.
I do not see any logical connection between anything you said.
You also didn't define "avg" even if it means "average", the average of two numbers isn't (x-y), but (x+y)/2.
I also see absolutely no connection to pi, nor have you stated ANY question, claim or problem.

>>11067414
So, what was your point?

>>11067414
What do you mean by this? Why wouldn't you want one?

>>11067420
My question is how tf do I solve a piecewise function with the fundamental theorem of calculus. Better?

>>11067355
How else would you decide whether or not [math]\mathbb R[/math] is isomorphic to [math]\mathbb R^2[/math] as a group ?

>>11067431
No, absolutely not.
You do not "solve" a function, especially do you not solve a piece wise function with the fundamental theorem of calculus, which by the way, is something ENTIRELY different to the mean value theorem.
I have absolutely no clue what that is supposed to mean or how it makes any logical sense.

Unless you state a specific problem where there is a specific answer and state it in mathematically well defined terms NOBODY here can help you.

>>11067439
Are you autistic?

>>11067447
Nobody here has any clue what you are saying.
You are literally speaking mathematical gibberish.

>>11067450
An easy "yes" would have been enough.

>>11067424
Because once you have some topology, you have the notion of topological span of a subset (ie. closure of the span), which is meaningful and actually useful in practice because you can find examples of topologically spanning sets in many vector spaces of interest, whereas I have yet to be shown a basis for an infinite dimensional complete normee vector space.

>>11067451
Junge, du bist echt behindert.
Allein schon das ich mir die Mühe mache deinen Unsinn in IRGENDEINE Form zu bringen war schon Zeit Verschwendung.

Du bist nicht autistisch genug für Mathe, werd Dachdecker, da bist du besser aufgehoben.

I have very bad memory it seems. I'm starting a PhD in maths soon, and I still forget formulas from first year Maths courses. How do I remember these? Flashcards? Anyone have experience with this. Should I go through textbooks and make notes?

I'm actually not terrible at maths besides this, but it makes me come across as a total brainlet.

>>11067457
>Unsinn

Junge, ich weis erst seit nem paar Tagen wie lineare und quadratische Mittelwerte funktionieren und die Englischen Begriffe sind zwar unterschiedlich, aber "mvt" und "ftfc" sehen aus wie nahezu das exakt gleiche und funktionieren grob nach dem selben schemata.
JEDER kann sehen was ich meine, nur weil du autist auf deinen "special autist club" beharrst macht das meine Frage nicht verwirrend.

>>11067462
>I'm starting a PhD in maths soon, and I still forget formulas from first year Maths courses. How do I remember these? Flashcards? Anyone have experience with this. Should I go through textbooks and make notes?
Why do you care to remember them?
You just need to know that they exist and where to look, seems like a waster of time to learn them by heart.

>>11067464
Aber weil du scheinbar Deutsch kannst, da, im Kontext, ohne was dazu zu schreiben, die c).

>>11067464
>JEDER kann sehen was ich meine
Nein, tatsächlich nicht.

> ich weis erst seit nem paar Tagen wie lineare und quadratische Mittelwerte funktionieren
Dann lass es dir von jemandem mit mehr Ahnung sagen das du totalen quatsch redest, weshalb ich auch die einzige person bin die dir antwortet, da der Rest sich nicht mal die Mühe macht dich zu verstehen.

>aber "mvt" und "ftfc" sehen aus wie nahezu das exakt gleiche und funktionieren grob nach dem selben schemata.
Sind KOMPLETT verschiedene Sachen übrigens die wenig, bis gar nichts, miteinander zu tun haben.

Du brauchst kein "mvt" und "ftc" um Dachdecker oder Klempner zu werden, deshalb rate ich dir werd Handwerker.

>>11067473
Wie fuehlt es sich an, mit 30 noch Jungfrau zu sein?

>>11067471
AH, du schaffst es also eine Frage zu produzieren.
Die wollen übrigens das du einfach integrierst, vermute ich.
b) und c) sind übrigens nicht wohl definiert, da "1/unendlich" nicht definiert ist und speziell nicht "1/unendlich*unendlich".

Welcher Idiot hat den diese Aufgaben verbrochen?

>>11067477
Bin 22, aber selbst da fühlt es sich sehr schlecht an.

>>11067465
Well as an example, I was doing a talk on Hidden Markov Models. Someone asked a question about a certain property. I tried to explain and someone had to correct me because I was completely wrong about the definition of the Markov property.

what does category theory have to do with trannies?

>>11067491
I mean, in the areas of your own research that is obviously different and it is even more different when you consider definitions.
I too, have quite a bad memory, but at least the definitions I have to work with regularly are quite present in my head.

If I were in your position I would probably set up some flashcard based learning program, into which I would enter every somewhat generally important definition or Theorem and do a couple of these flash cards each day.

Ok so I have an exercise dealing with compact spaces, open balls and sets. I'm struggling with the notation. For all [math]x\in E[/math] there is some specific [math]r[/math] dependent of [math]x[/math] such as [math]B(x,r)\subset E[/math]. Now I want to express the union of all such balls, how could I clearly say that? Is [math]\bigcup_{x_i\in E}B(x_i, r_i)[/math] an acceptable notation? It's awkward as the subscript [math]i[/math] is just a dummy symbol to remind us that [math]r_i[/math] depends on [math]x_i[/math].

What's a good book on proofs?

>>11067498
https://www.twitter.com/category_fury

>>11067597
Here is a good free book for beginners looking at the usual methods to prove something : https://www.people.vcu.edu/~rhammack/BookOfProof/ (disclaimer: it's the only book about proofs I read). "How to prove it" is also a popular book on the subject (also for beginners).

If you had a midterm that you bombed like <40% what would be your best plan of attack for the future? Go back and re-learn/study from the beginning and fall behind on current concepts or learn current concepts and just never get around to fully understanding the initial concepts?

Totally depends whether the second part of your course is highly dependent on the deep understanding of the beginning of the course.

>>11067608
Thanks.

>>11067597
I think it's a bad idea to read a book solely about proofs. It's better to learn proof techniques in the context of something else, like calc or linear algebra.

>>11067594
The notation is correct, but in this case, it's "every x_i in E", which means, the whole of the space E
Maybe you meant just a sequence of x_i, in which case, "union of x_i" or even "union over the i" would be enough for me

>>11065894
>Not that great for teaching yourself the super hard questions and techniques to solve shit
What would you recommend in that case?

>>11067013
Dankeschön!

Ah, I see. f is both the function and the chart.
Absolute genius.

>>11067289 not a problem
>>11065096

>>11067594
That's kinda how I do it. I usually specify that for all x in some open set, there exists ε(x) s.t the ball with that radius around x is contained in the open set

If you assume the axiom of choice is in the category of axioms, can you create a choice function that picks the axiom of choice?

>>11067315
Circular logic is circular.
Non-measurable sets are another unintuitive consequence of choice. Your intuition about uncountable sets is shaped by how they behave under choice. You just learnt to accept the bad consequences of choice.

>>11067701
category theory was invented precisely to answer these sort of questions

>>11067701
https://ahilado.wordpress.com/2016/12/13/even-more-category-theory-the-elementary-topos/

>>11067701
This isn't math, it's just masturbation at this point

>>11067412
>strong theories like Z

>>11067452
Adding the condition of local convexity, you can leverage the spectral theorem to produce an ONB. See e.g. [math]N[/math]-representation theorem for tempered distributions.

>>11067452
Is the existence of such a (Schauder)Basis probable without AC in the situations in which you would want to use them?

At least in some cases they can be constructed, but in all useful cases?

>>11067925
We can't even necessarily give a Schauder basis for a separable Banach space lmao.

>>11067936
>We can't even necessarily give a Schauder basis for a separable Banach space lmao.
That doesn't answer my question in the slightest, it seems in fact entirely irrelevant.

>>11067946
>can we usually do this?
>not really, we can't do it in this one basic case, for example
>that reply is completely irrelevant to my question
Are you a dunce?

speak german to me

>>11067634
The final will probably be half and half. So even if you get 100% on the second half and 40% on the first half, you'll only wind up with a 70% (100%/2 + 40%/2).

So I'd start back over from the beginning and do any practice tests/homework you can do. Talk to the professor about what they recommend.

I remember in high school, algebra was all about proofs, using/memorizing different theorems, and just overall doing things in a very rigid step-by-step process.

Is that not the case anymore? I'm taking college algebra and it's just "do it however you want".

>>11067599
he cute

needs to clean his mirror though

>>11066760

This isn't true. I correctly identified the foolish image choice by the OP, here >>11065430 . What followed were various incorrect "nuh-uhs" of which yours, being rhetorical and actually cogent, is the best so far (though still wrong).

>>11067796
kek, granted

>>11065437
>>11065437
Actually fun question, a first on this general.
We begin by treating the analogous problem in dimension 2, ie. we pick a point [math]P[/math] in the interior of an ellipse of [math]\mathbb R^2[/math] with its usual inner product.
Since the problem is invariant under translation, we may assume that our ellipse is centered at the origin, and therefore has an equation of the form [math]E = \{z \in \mathbb R^2, q(z) = 1\}[/math], with [math]q[/math] a Euclidean norm on [math]\mathbb R^2[/math].
Then, our assumption on [math]P[/math] amounts to assuming [math]q(P) < 1[/math].
First, we reduce to the case where [math]P = O[/math].
If [math]u[/math] is a nonzero vector, which (rescaling if necessary) satisfies [math]q(u) = 1[/math], the intersection points of the line through [math]P[/math] directed by [math]u[/math] with [math]E[/math] are of the form [math]A_i = P + t_i u[/math], where [math]t_1, t_2[/math] are the two solutions of the quadratic equation [math]q(P+tu) = 1[/math], which rewrites as [math]t^2 + 2t\langle P, u \rangle_q + q(P) = 1[/math].
In particular, they satisfy [math]t_1 t_2 = q(P) - 1[/math].
Then, we have [math]\frac{1}{PA_1 \cdot PA_2} = \frac{1}{|t_1t_2|||u||^2} = \frac{1}{(1-q(P))||u||^2}[/math].
Applying the same reasoning to the orthogonal line through [math]P[/math], we get [eqn]\frac{1}{PA_1 \cdot PA_2} + \frac{1}{PB_1 \cdot PB_2} = \frac{1}{(1-q(P))}\left(\frac{1}{||u||^2} + \frac{1}{||v||^2}\right),[/eqn]
where [math]u[/math] and [math]v[/math] are two orthogonal vectors lying on the ellipse.
Next, we introduce some coordinates to prove that the quantity between brackets is independent of [math](u,v)[/math].
(cont.)

>>11068127
By the spectral theorem, we know that we may choose our orthonormal coordinate system on [math]\mathbb R^2[/math] in such a way that, for any [math](x,y) \in \mathbb R^2[/math], we have [math]q(x_1, x_2) = \frac{x_1^2}{a_1^2} + \frac{x_2^2}{a_2^2}[/math], for some [math]a_1, a_2 > 0[/math].
Then, if [math]u = (u_1, u_2)[/math] is on [math]E[/math], we must have [math]v = \xi(-u_2, u_1)[/math], where [math]\xi[/math] satisfies [math]q(v) =1[/math], ie. [math]\xi \left(\frac{u_2^2}{a_1^2} + \frac{u_1^2}{a_2^2}\rangle) = 1[/math].
Then, our quantity rewrites as
[eqn]\frac{1}{||u||^2} + \frac{1}{||v||^2} = \frac{1}{u_1^2 + u_2^2} \left(1 + \frac{1}{\xi^2}\right) = \frac{1}{u_1^2 + u_2^2}\left(\frac{u_1^2}{a_1^2} + \frac{u_2^2}{a_2^2} + \frac{u_2^2}{a_1^2} + \frac{u_1^2}{a_2^2}\right) = \frac{1}{a_1^2} + \frac{1}{a_2^2}.[/eqn]
This proves the independence on the pair of orthogonal lines in dimension 2.
Next, we see how this directly implies the case of dimension 3.

>>11068127
By the spectral theorem, we know that we may choose our orthonormal coordinate system on [math]\mathbb R^2[/math] in such a way that, for any [math](x,y) \in \mathbb R^2[/math], we have [math]q(x_1, x_2) = \frac{x_1^2}{a_1^2} + \frac{x_2^2}{a_2^2}[/math], for some [math]a_1, a_2 > 0[/math].
Then, if [math]u = (u_1, u_2)[/math] is on [math]E[/math], we must have [math]v = \xi(-u_2, u_1)[/math], where [math]\xi[/math] satisfies [math]q(v) =1[/math], ie. [math]\xi \left(\frac{u_2^2}{a_1^2} + \frac{u_1^2}{a_2^2}\right) = 1[/math].
Then, our quantity rewrites as
[eqn]\frac{1}{||u||^2} + \frac{1}{||v||^2} = \frac{1}{u_1^2 + u_2^2} \left(1 + \frac{1}{\xi^2}\right) = \frac{1}{u_1^2 + u_2^2}\left(\frac{u_1^2}{a_1^2} + \frac{u_2^2}{a_2^2} + \frac{u_2^2}{a_1^2} + \frac{u_1^2}{a_2^2}\right) = \frac{1}{a_1^2} + \frac{1}{a_2^2}.[/eqn]
This proves the independence on the pair of orthogonal lines in dimension 2.
Next, we see how this directly implies the case of dimension 3.
(cont.)

Question to MathPhys guy. Somewhat of a vague question, but to what extent does physics deal with non-separable spaces? It seems intuitive that anything with physical significance ought to be separable.

Is abstract algebra hard or am I just a fucking retard? I'm a civil eng major taking intro to abstract for a math minor and it's the most fucking absurd, bizarre, and nonsensical class I've taken in my life. Does this shit come easily to you guys or something? Do you just have to be born with an autistic predisposition to being good at math to get it? Did anyone else struggle trying to figure this shit out?

>>11067982
>>can we usually do this?
I asked about useful cases.
Whether or not it is true in general couldn't be less relevant.

I asked whether there are useful cases where you need AC to get a Schauder Basis.
And you replied that in general it isn't possible, which has nothing to do with my question, except reaffirming it's validity.

>>11068163
Banach spaces are the most useful cases here so he did answer your question.

>>11068152
Algebra (at least elementary algebra) is one of those things that flips like a switch from what-the-fuck to everything-is-obvious without much of a gap between them. Most people go through both phases, and the ones that don't are usually cheating by having prior exposure through contests or a number theory class that spent some time on the integers mod N or something.

I think for most people (extrapolating my own experience here) the difficulty with algebra is that they overthink it at first. Once you get used to them, proofs in basic algebra almost all work the same; there's only one thing you could really do that makes any sense, you write it out, and it works. The part that takes practice is getting your brain into the spot where it's obvious that there's only one thing to try.

>>11068152
Can you explain what you're finding hard?

I hate it when people come over here complaining about how bad theyre doing and they don't even say whats wrong. How can we help you if you're making it as hard as possible for us?

And yes, abstract algebra is hard. There is no innate intuition to why some things behave the way they do. But there is rhyme and reason as to why some things work and some don't. The problem, as with many other subjects in math, is that you learn the formalism and theorems before you learn of the objects, when in reality, the study of the objects precede their later abstractions.

Galois didn't study separable and normal field extensions, or normal subgroups, before he found their relationship with polynomial roots. Yet you only study what Galois proved after 3 courses of pure algebra.

Of course, there is some sense to it. Ultimately, you could spend a year studying the background and have a solid understanding of examples, or you could study the theory in a month and get the gist of it. So it's more efficient if you have a further goal.

>>11068152
>Is abstract algebra hard or am I just a fucking retard?
sometimes
> I'm a civil eng major taking intro to abstract for a math minor and it's the most fucking absurd, bizarre, and nonsensical class I've taken in my life. Does this shit come easily to you guys or something?
no
>Do you just have to be born with an autistic predisposition to being good at math to get it?
no
>Did anyone else struggle trying to figure this shit out?
sometimes

Von Neumann once said: "Young man, in mathematics you don't understand things. You just get used to them." Apply this to your course.

>>11068195
>>11068152
Also just to add to that, most things in beginner algebra are ultimately just really obvious once it clicks, as the other anon said. The first isomorphism theorem is confusing until eventually you see that there is no other sane alternative, and it really just follows from the definitions. But you have to fight with it until it is obvious.

There's a joke that goes:

>Two mathematicians are discussing a theorem. The first mathematician says that the theorem is “trivial”. In response to the other’s request for an explanation, he then proceeds with two hours of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial.

>>11068139
Now, we go back to our original setting, with [math]E[/math] an ellipsoid in [math]\mathbb R^3[/math] and [math]P[/math] an interior point of [math]E[/math]. Then, for each triplet [math](L_A, L_B, L_C)[/math] of orthogonal lines passing through [math]P[/math], we set [eqn]F(L_A, L_B, L_C) = \frac{1}{PA_1 \cdot PA_2} + \frac{1}{PB_1 \cdot PB_2} + \frac{1}{PC_1 \cdot PC_2},[/eqn]
where [math]A_1[/math] and [math]A_2[/math] (resp. [math]B_1[/math] and [math]B_2[/math], [math]C_1[/math] and [math]C_2[/math]) are the intersection points of [math]L_A[/math] (resp. [math]L_B[/math], [math]L_C[/math]) with [math]E[/math].
Then, note that, if we fix [math]L_A[/math], then it follows from all of the above that [math]F(L_A, \cdot, \cdot)[/math] does not depend on the pair [math](L_B, L_C)[/math], as long as both lines are perpendicular, and both perpendicular to [math]L_A[/math].
Indeed, in that case [math]P[/math] is an interior point of the ellipse [math]E \cap \mathrm{AffineSpan}(L_B, L_C)[/math] and the quantity [math]\frac{1}{PB_1 \cdot PB_2} + \frac{1}{PC_1 \cdot PC_2}[/math] is precisely the one that we proved to be constant in the previous section.
Moreover, since [math]F[/math] is invariant under permutation of the arguments, the same applies if we fix [math]L_B[/math] or [math]L_C[/math] and vary the other two lines.

>>11068152
>Does this shit come easily to you guys or something?
Yeah, I've picked up the capacity to derive results from abstract axioms with no geometrical interpretation from high school physics tests with formulas in the back.
Absolute dogshit at number theory, tho.
>>11068201
The weird part of that von Neumann story is that the method of characteristics is simple as shit as long as you understand that derivative everywhere zero implies constant function.

>>11068224
For any pair of orthogonal lines through [math]P[/math], say [math](L, L')[/math], write [math]L \wedge L'[/math] for the perpendicular to the affine plane spanned by [math]L[/math] and [math]L'[/math] passing through [math]P[/math].
Now, let [math](L_A, L_B, L_C)[/math] and [math](L_{A'}, L_{B'}, L_{C'})[/math] be two triplets of orthogonal lines.
Then, if [math]L_A = L_{A'}[/math], we are done by the above argument.
Otherwise, we write
[eqn]F(L_A, L_B, L_C) = F(L_A, L_A\wedge L_{A'}, L_A\wedge(L_A \wedge L_{A'})) = F(L_{A'}, L_A \wedge L_{A'}, L_A' \wedge (L_A \wedge L_{A'}) ) = F(L_{A'}, L_{B'}, L_{C'}[/eqn]

I'm going to fail, what can I do now

>>11068228

mirin the latech

>>11068230
bring some kneepads to office hours

>>11068230
mitigate the damage as much as possible. or >>11068241

>>11068228
Good job. Nice to see this thread isn't completely full of brainlets

>>11068012
nein

>>11068193
Modular arithmetic is one of the only things that I kind of understand in the class. Last lecture, we started going over polynomial rings and we discussed how deg(f) + deg(g) = deg(f*g) only if R is an integral domain. When we went through the proof, I understood the example where he used R = Z6 and was the only person actively answering his questions.
>>11068195
Well, proofs and logic is the hardest thing for me because I never took discrete, which is apparently highly recommended. Then there's just some basic ideas that aren't really registering with me, like zero divisors, which is apparently a huge thing and pretty simple, but it's just not really clicking with me. We just wrapped up a section on homomorphisms, and I still don't really know what the fuck that means. To my knowledge, a homomorphism is just a map from one ring or some shit to another, and it's isomorphic if it's a surjection and injection, and if it's not both it's still a homomorphism? I guess.
>>11068201
Thanks, I'll remember that.
>>11068210
I think one of the biggest problems is that since it's not a class for my major, and it's just a minor I'm doing, I'm not really applying myself and studying as hard as I should be for the class, i.e. I'm not fighting with it enough. Honestly the class kinda feels like an art movie; I sit down, smile and nod my head, and when I leave I have more fucking questions than I did before. And whenever I try talking to my batshit crazy Hungarian professor after class or during office hours, we spend 30 minutes and he shows me random shit in graph theory that I don't really understand.

>>11068012
wie geht's?

Post your top 5 of all time.
Gromov, Riemann, Poincaré, Leray, Bergman.

>>11068282
>Well, proofs and logic i...

Rings are defined abstractly, it is a set that follows a couple of rules. The rules it must follow are very similar to the basic properties of the integers (except commutativity of multiplication). However, since the definition is abstract, it actually has many more objects other than the integers as examples. A big one is matrices. Since you have so many objects that follow the rules of rings, you 'accidentally' include ones that are very abnormal to intuition. Such things as zero divisors exist in matrices: two non-zero matrices can multiply to create a zero matrix, something that can't happen in the integers or the real numbers. This is a disadvantageous property to have in rings, so sometimes we want to exclude the study of these rings and therefore look at integral domains.

Homomorphisms are essentially just functions between rings that preserve the ring structure. For example, if I have a function [math]f:\mathbb Z\to \mathbb Z[/math], then suppose that [math]f(2)=3, f(4)=5[/math]. Then [math]f(2+2)=5[/math] but [math]f(2)+f(2)=3+3 = 6[/math]. Therefore, we have 'broken' the ring structure of the image, and the whole point of the target of the function being a ring is completely irrelevant then.

If a homomorphism is injective, what it tells you is that the ring of the source gets 'transported' into the ring of the target, such that the image and the source can be identified.

For example, consider [math]f:\mathbb Z/2\mathbb Z\to \mathbb Z/6\mathbb Z[/math] where [math]f(0)=0, f(1)=3[/math]. Note that [math]1+1=0[/math] in [math]\mathbb Z/2\mathbb Z[/math] and [math]3+3=0[/math] in [math]\mathbb Z/6\mathbb Z[/math] . Also [math]0=f(0)=f(1+1)=f(1)+f(1)=3+3 =0[/math], so it is well defined and injective. Notice how [math]\mathbb Z/2\mathbb Z[/math] has been mapped into the subset [math]\{0,3\}\subset \mathbb Z/6\mathbb Z[/math], and that subset has a ring structure that behaves exactly like the domain of [math]f[/math]

>>11068342
Surjection just means that the homomorphism has hit all of the target ring.

If a homomorphism is an injection and a surjection, then, in words, the function maps the domain into the target creating a copy of itself in it (injection), however, it has also hit all the elements in the target (surjection), and further, the ring structure of the domain is preserved in this image (homomorphism).

This essentially means that the two objects are basically the exact same ring, however the labels of the elements might be different, but it's just a label, so it's irrelevant - they are the same ring for all intents and purposes. So it's an isomorphism.

>>11066981
>>The undergrad category tranny pseud is at it again
Who are you quoting?

>>11068336
Cauchy, Cantor, Euler, Lebesgue (get dabbed on Riemann), Gödel

Can someone explain what is happening in the 2nd step?

They take the ln of both sides and the sqrt(x) comes down. Then they differentiate implicitly. They then appear to use the product rule?

When the sqrt(x) moves down, this is not because of the power rule (since the exponent is variable) but because of log rules that exponent can move down multiplying the ln as a scalar quantity correct?

>>11068382
you are correct

>>11068382
I'm not sure what you're confused about. You've pretty much explained the entire pic correctly.

>>11068382
>They then appear to use the product rule?
Yes.

>When the sqrt(x) moves down, this is not because of the power rule (since the exponent is variable) but because of log rules that exponent can move down multiplying the ln as a scalar quantity correct?
No, it is because of the power rule (exponent is not variable, sqrt(x) = x^(1/2)):
d/dx(sqrt(x)ln(x)
= sqrt(x)d/dx(ln(x)) + ln(x)d/dx(sqrt(x))
= sqrt(x)d/dx(ln(x)) + ln(x)d/dx(x^(1/2))
= sqrt(x)d/dx(ln(x)) + ln(x)(1/2)x^(-1/2)
= sqrt(x)d/dx(ln(x)) + ln(x)(1/2)/sqrt(x)

>>11068342
Alright yeah, I remember we did zero divisors with matrices, and like went over how they can be left or right zero divisors. I think what's fucking me over is that I didn't really think that they were specific to examples like matrices, and that clears my hangups with integral domains, too; so basically, an integral domain is just a ring that has no zero divisors. Are there any other examples of things like matrices that can yield zero divisors that I'd be able to understand at a basic level? Just out of curiosity.
I'm understanding where you're headed with homomorphisms, but one problem I'm not really getting is that my professor never really told us what an image is. What exactly does it mean when you say that you can identify the image in an injection? Am I right in assuming that the image is just the "realm" (for lack of a better word) that you're going into? If I had to put it into my own words, I'd say that in an injection, when you go from one ring to another, the function from the original ring is equal to the function in the target ring. That's kinda what I get from your example, but that just feels like I'm getting the wrong thing from it. I think I'm just not getting your notation because we haven't done anything with stuff like z/2z, every map we've done so far is just like z -> z6 and shit like that, for example. I don't get what z/2z means; is it supposed to be read like "Z divides 2Z" or something like that?

>>11068387
>>11068395
thanks

>>11068396
I mismarked the picture, the yellow wasn't supposed to be highlighted.

>>11068456
>>11068456
> Are there any other examples of things like matrices that can yield zero divisors that I'd be able to understand at a basic level? Just out of curiosity.

For deep reasons, giving examples that aren't matrices is a bit hard. However, there is a nice construction that does yield these rings, called the direct sum:

Let [math]R,S[/math] be any two rings. Construct a ring [math]R\oplus S[/math] in the following way:

Every element of [math]R\oplus S[/math] is a pair [math](r,s)[/math] where [math]r\in R, s\in S[/math].

Addition is defined as: [math](r,s)+(r',s')=(r+r',s+s')[/math]

Multiplication is defined similarly: [math](r,s)\cdot (r',s')=(rr',ss')[/math]

You can check this forms a ring, and indeed, any element [math](r,0)[/math] is a zero divisor in this way, since [math](r,0)\cdot(0,s)= (0,0)[/math].


The image of a function is the set of elements of the target that the function 'hits'. When you have an injection, every element is sent to precisely one element in the target. Therefore if you only know the image and that the function is an injection, you can 'recreate' the domain ring from only that information. By obvious reasons, every function is surjective on its image, so when I say you can identify the domain with the image of an injection, I literally mean that the domain is isomorphic to its image, which is a subring of the target ring.

Also, the notation [math]\mathbb Z/2\mathbb Z[/math] is just the integers modulo [math]2[/math], which I assume you know about them since you mentioned them in a previous post - perhaps you saw different notation.

>>11068375
>get dabbed on Riemann
Lol, the Riemann integral was still very good for its time. You can't expect the first definition to be perfect. Plus, Riemann's total contributions exceed Lebesgue's so whatever.

>>11068497
Holy shit yeah, I saw something about that in the book but we didn't do it in class. Thanks, that's starting to make a lot more sense now.
And alright, putting it that way makes it easier to understand, and it really clears up why he kept referring to the as one-to-one functions. And yeah, we use different notation, I don't go on this board much so I don't know how to do all that math symbol shit but the notation we use is basically Z sub 2 for integers mod 2. I was a little confused by that. That really clears up the example, and I
Thanks a ton, I think I understood more from that than I have in the fucking class the past few weeks. Do you happen to know any good youtube channels that I can reference in the future for if I need something cleared up? Khan academy usually has a lot of shit for my other courses but I can't find an abstract algebra series by them, unless it's buried in their general algebra playlist somewhere

>>11068582
I can't really recommend any channel since I've never seen a yt series on algebra but I'm sure there's plenty of lecture courses online - by this I dont mean a youtuber, I mean a proper lecturer giving a lesson

>>11068369
im not quoting anybody you disgusting degenerate freak. The day you off yourself is the day this world gets a little bit better. you do not understand how tempting it is to wrangle your scrony little necks when I here one you faggots trying to talk about abstracted thing you dont understand.

>>11068821
>im not quoting anybody you disgusting degenerate freak. The day you off yourself is the day this world gets a little bit better. you do not understand how tempting it is to wrangle your scrony little necks when I here one you faggots trying to talk about abstracted thing you dont understand.
Are you alright?

>stuck on obscure logic problem
>sick for the lecture we covered the concept behind it
>only fucking one textbook which even mentions this shit and it's the one we're assigned, which i can't make sense of for this problem
>have no clue how to solve it
>no online resources whatsoever, no recordings of other lectures on the topic, nothing
>my final last hope, /sci/
>come to /mg/
>"no logic allowed edition"
https://youtu.be/EQ3G0oS-OWY?t=191

Does anyone have a recommendation for a nice fluid- and/or aero-dynamics textbook? I do not need any additional mathematical "background" in PDEs and DG, so it can be an advanced text.

>>11069142
Last thread some guy fight about LEM, but logic is OK.

>>11069176
https://www.springer.com/gp/book/9780387949475

https://www.cambridge.org/core/books/vorticity-and-incompressible-flow/393C35E544EDD0711CAA7F7AB05D7432

what the fuck is fucking vector spaces and bases and shit I don’t understand this FUCK

>>11069268
Thanks!

>>11067705
But none of these things are unintuitive. They only are because you're not intuiting it. They're just not things you can write down. They're still perfectly intuitive.
Sure, my intuition is shaped by choice, but that's the right kind of intuition to have. The wrong kind would be to reject anything which isn't completely countable in nature as inconceivable - the continuum is one of the most natural, intuitive objects to construct and you're essentially opposed to studying it in the most intuitive ways which could ever come to someone's mind.

>>11069270
A vector space is a set of "vectors" equipped with several properties. (https://en.wikipedia.org/wiki/Vector_space#Definition, http://www.math.toronto.edu/gscott/WhatVS.pdf page 3, http://www.math.niu.edu/~beachy/courses/240/06spring/vectorspace.html))

A basis is a subset of a vector space such that the elements of said basis can be linearly combined to form each element of the vector space (is generator) uniquely (linear independence). So if [math] a_1 v_1 + a_2 v_2 +...+a_n v_n = u \in V [/math] and [math]b_1 v_1 + b_2 v_2 +...+b_n v_n = u [/math] then the sequences are equal ie: [math] a_n= b_n \hspace{0.1cm} \forall n [/math]. Also, [math] \forall u \in V, \exists v_n,a_n:a_1 v_1 + a_2 v_2 +...+a_n v_n = u [/math] with [math] v_n [/math] being the vectors of your basis.

>>11067359
you're acting like it's not an obvious and immediately clear fact that every vector space morally should have a basis
just keep picking linearly independent vectors infinitely often until you run out of linearly independent vectors to pick. it's so fucking obvious. jesus christ.

>>11068282
>it was HIGHLY RECOMMENDED to take discrete but i just didn't do that
>wah wah wah it's so hard
congratulations, dumbest post i've read all week. that's a fucking prerequisite you moron.

>>11069343
>morally

>>11069441
you're not a mathematician and you never will be one

if I have [1+(1/x)] * [1/(x+lnx)], don't I distribute each term? I got an answer of 1/(x^2+lnx), but it's wrong. Do you have to distribute the entire (1+1/x) expression to the numerator?

On the plane R^2, consider the monotone polynomial
y(x) := x^n
for some positive integer n And evaluated on positive arguments X. I I.e. that's a monotone steep curve on the upper right quadrant of R^2. The points parameterized by x are (x, x^n).

I'm having a hard time finding out the polar parameterization r(a) such that that points are (r(a), a), where a is the angle between 0° and 90°.
That is to say, express the distance r(a) of a point on the curve from the origin in terms of the angle (or, equivalently, in terms of the slope of of a line going through the origin and intersecting the curve.).

>>11069519
>expression
What do you mean?

>>11069343
>morally
Math isn't about morality. It is about discovering theorems with a given set of axioms.

>>11069540
x = r cos(a)
y = r sin(a)
y = x^n
okay, try working it out yourself from here.


if you want to see the answer, here it is:
r sin(a) = r^n (cos(a))^n
r^(n-1) = sin(a)/(cos(a))^n
r^(n-1) = tan(a)(sec(a))^(n-1)
r = (tan(a))^(1/(n-1)) * sec(a)
that should work

>>11069666
>math isn't about morality
>post number
yeah right math satan

>>11069666
Take your meds, Hilbert

>>11068840
Just kill yourself already tranny

Verbitsky's website has tests he's given to his students. Are these normal? Because they're 90% memorization.
http://verbit.ru/IMPA/TOP-2018/

>>11069671
Thanks

>>11069337
>But none of these things are unintuitive. They only are because you're not intuiting it. They're just not things you can write down. They're still perfectly intuitive.
But none of these things are intuitive. They only are because you forced yourself to. They're not things you can write down. There is no intuition.

>Sure, my intuition is shaped by choice, but that's the right kind of intuition to have.
Because you say so? Because you were taught this way?

>The wrong kind would be to reject anything which isn't completely countable in nature as inconceivable - the continuum is one of the most natural, intuitive objects to construct and you're essentially opposed to studying it in the most intuitive ways which could ever come to someone's mind.
Not how the world without choice works.

>>11069142
Ask a classmate for their notes. You do have friends right anon?

>>11069980
no.

>>11069337
Let P be the proposition
P := "this sentence is false"

Let p be a natural number such that
p is 3 if P is true and p is 7 is p is false

Let Q3 be the proposition asserting that p is 3, i.e.
Q3 := (p = 3)

>I defined p to be a natural number, I can perfectly Intuit it
>Obviously we must adopt LEM. For any proposition, they are either true or false. It's clearly evident. A precondition of mind.
>Clearly "Q3 or not Q3" is true. Either Q3 is true or Q3 is false. In the Platonic realm, p is 3 (or maybe 7, I'm not sure. But it has one, clearly. One of those, per definition). Evidently p is a number, independent of us. I defined it so. A number without a value, who came up with that!
>Clearly all vector spaces have bases, even if I can provably not state any basis for it.

(Replace the impredicative sentence P with any real that has its digits defined in terms of an halting problem)

>>11069983
>the proposition P := "this sentence is false"
No such thing.

>>11069983
>For any proposition, they are either true or false. It's clearly evident.
[Ontological basis required]

>>11069996
See my last comment.

E.g. replace P by
"the 100th digit in base 10 of the Chaintlin constant is 0"

https://en.m.wikipedia.org/wiki/Chaitin%27s_constant

>>11069983
>In the Platonic realm,
What did he mean by this?

Hi, got something to make your heads hurt.
X+1=-x^2
x^2 + x + 1 = 0
(x certainly isn't 0)
x + 1 + 1/x = 0
(from the original statement)
-x^2 + 1/x = 0
-x^3 + 1 = 0
x = 1

Where's the problem?

>>11070289
>X+1=-x^2
There is no such x.

>>11070399
Doesn't explain why the steps don't hold, does it? Besides.. complex

>>11070289
You made no mistake, except that you introduced a third root (when substituting the equation again) and solved the last equation wrong.
-x^3+1=0 has three solution in the complex plane, you introduced 1 artificially, but both of the other solutions are actually correct solutions to the initial problem.

>>11069983
>Platonic realm
Cringe

>>11067705
If you also have to learn to accept the bad consequences of not accepting choice
>Existence of an infinite set with no injection of N into it
>Some sets can be partitioned into subsets which, taken as a collection, will have a greater cardinality than the set of which they partition
And much more

>>11070448
Why isn't 1 then?

>>11070432
Double check using the quadratic formula

>>11070516
>Why isn't 1 then?
What do you mean? That isn't a complete English sentence.
x^2=1 has two solutions
x^3=x has three solution, although obviously all I did was multiplying x.

New thread

>>11070543
>>11070543
>>11070543

>>11070528
Yes, but 1 clerly doesn't solve the original equation

>>11070550
And zero certainly doesn't solve x^2=1, but it clearly solves x^3=x.
Where is the surprise? Multiplying by x isn't an equivalence relation and what you did is just something more sneaky then that, but ultimately just as unsurprising.

>>11070550
In fact all your manipulations are just equivalent to multiplying with (x-1), which makes it extremely unsurprising that you introduce 1 as an additional solution.

http://mathonline.wikidot.com/the-cancellation-law-for-groups
Why is this valid when the supposed identity e isn't defined inside the group G?

>>11067359
Did you just actually, seriously imply that vector spaces "exist"?

>>11067486
Fuck jetzt fuehl ich mich scheiße

Tschuldige Bro, manchmal kommt der Hauptschul Assi aus mir raus :(

Verbrochen hat die Aufgabe meine Medizintechnik Professorin die gleichzeitig Ufer Mathe zustaendig ist.

>>11070692
fuer*

>>11070692
Bei b) kann ich dir unironisch nicht sagen was gemeint sein soll.
Selbst eine "rigorose" Grenzwert Betrachtung hilft halt nicht wirklich weiter und ich glaube nicht das es eine sinnvolle Definition des "linearen/quadratischen Mittelwerts" gibt die hier anwendbar wäre (das Problem ist halt, selbst wenn man den Grenzwert betrachtet ist es der limes einer Periodischen Funktion und existiert demzufolge nicht).
Deine Professorin sollte sich mal eine Jungfrau aus der Mathematik schnappen die sich ihre Aufgaben durchließt, denn selbst ich (als mittlerweile Universität zertifizierte Mathe Jungfrau) kann der Frage keinen Sinn abgewinnen.

Glaube aber auch nicht das irgendetwas davon mit dem "Mittelwertsatz" zu tun hat, der sowieso nur auf Intervallen endlicher Länge anwendbar ist.

>>11070289
(x+1)^2 is not x^2+x+1
(x+1)^2+x^2 is not x^2+x+1 either.

>>11071068
Nothing you say makes any sense.
His algebra is perfectly fine, see >>11070448

>>11064771
Based Buffy poster

>>11071071
Shit sorry thought it was another "square it and 1=0 wow" post

the pursuit of advanced mathematical understanding is a meaningless trivial endeavor.

>>11071906
>>11072096

>>11072098
Some people are more “other-directed” than others, and therefore will more readily attach importance to a surrogate activity simply because the people around them treat it as important or because society tells them it is important. That is why some people get very serious about essentially trivial activities such as sports,or bridge, or chess, or arcane scholarly pursuits, whereas others who are more clear-sighted never see these things as anything but the surrogate activities that they are, and consequently never attach enough importance to them to satisfy their need for the power process in that way