/sci/ - Science & Math

Threadly reminder to work with physicists.

>>9459278
>Threadly reminder to work with physicists.
This.

>>9459426
>axiom of choice is required to prove every vector space has a basis
You might as well just take "every vector space has a basis" as an axiom then, since it's isomorphic to the axiom of choice as objects in the category of axioms.

>>9459431
>Category of axioms
What would that be?

What's the mathematical definition of "economy"?

How come when you run code multiple times on a computer it will take a different amount of time each run?

https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/

>Polymath proposal: upper bounding the de Bruijn-Newman constant

>Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant [math]\Lambda[/math]. The purpose of this post is to describe the proposal and discuss the scope and parameters of the project.

>>9459684
are you using an OS?

>>9459684
There are several issues: A modern personal computer switches all 10-100 ms the process ("context switch" see wiki), which makes it not easy to measure execution time. Then it could be that some caching mechanism alters programm execution time. Maybe the OS has to do some jobs and interrupts you process. Etc.

What does /mg/ think of this book collection? I tried to keep it one-book-per-subject, covering essential stuff from the first years in uni with a rigorous, proof-heavy approach

>Logic: The Laws of Truth
>Enderton's Set Theory
>Herstein's Topics in Algebra
>Landau's Foundations of Analysis
>Spivak's Calculus
>baby rudin
>Hardy, Littlewood & Polya - Inequalities
>Meresev's Fundamental Concepts in Algebra
>Apostol's Analytical Number Theory
>Munkres' Topology
>Hoffman & Kunze - Linear Algebra
>Edwin Moise's Geometry

I'd also like some books that'd fit this list on probabilities (kolmogorov books maybe?), complex analysis, calculus on manifolds and number theory (I know about Hardy's but I don't really like it)

and math books discussion general I guess

What does /sci/ think of this book collection? I tried to keep it one-book-per-subject, covering essential stuff from the first years in uni with a rigorous, proof-heavy approach

>Logic: The Laws of Truth
>Enderton's Set Theory
>Herstein's Topics in Algebra
>Landau's Foundations of Analysis
>Spivak's Calculus
>baby rudin
>Hardy, Littlewood & Polya - Inequalities
>Meresev's Fundamental Concepts in Algebra
>Apostol's Analytical Number Theory
>Munkres' Topology
>Hoffman & Kunze - Linear Algebra
>Edwin Moise's Geometry

I'd also like some books that'd fit this list on probabilities (kolmogorov books maybe?), complex analysis, calculus on manifolds and number theory (I know about Hardy's but I don't really like it)

also math books discussion general I guess

>>9459925
>>baby rudin
Rudin is a meme.

Someone sell me on HoTT

>>9459928

Rudin is a meme is a meme

>>9460240
For all X, "X is a meme" is a meme

>>9460185
Well do you want some of that HoTT Coq?

Should I learn measure theory before or after analysis?

>>9460629
I'd like to see you do it before.

I'd just like to interject for a moment. What you’re referring to as Bayesian statistics, is in fact, nonsense, or as I’ve recently taken to calling it, the inverse probability.

The inverse probability is not a method unto itself, but rather another work or fiction made useful by the idiots that proclaim to know some prior probability, these idiots are in fact worse than those that reject the axiom of countable additivity and instead embrace the axiom of finite additivity.

You see, Bayes identified the problem, provided the solution and let the whole idea die with him as he too understood the inanity of the inverse probability. Laplace on the other hand was too arrogant and decided to pursue these trivialities.

The only question that you should ask yourself related to Bayesians is as follows. Bayesians: knaves or fools?

Gonna need a swift and definitive gestallt on Duality, in particular dual vector spaces / dual basis please.

>>9460718
>these idiots are in fact worse than those that reject the axiom of countable additivity and instead embrace the axiom of finite additivity.
What are your preferred axioms?

>>9460271
Is this statement a meme though?

>>9459925
classical introduction to modern number theory, Rosen and Ireland
Lee's smooth manifolds
ahlfors' complex analysis

>>9460771
this statement is undecidable in the category of all axioms

>>9459925
Foundational stuff (set theory, formal construction of reals, etc.) is not worth reading unless you're an autist who likes it.
Most of the mathematicians in your department probably know the Peano axioms exist but I bet at most a handful could actually describe them to you with any accuracy, because nobody gives a single shit except to know that what they know is obvious isn't wrong.

>Intuitionistic logic is weaker than classical logic. Each theorem of intuitionistic logic is a theorem in classical logic.
LMAO

>>9459308
We need a new version of this that is actually readable.

>>9460883
>We need a new version of this that is actually readable.
What can you not read?

>>9460883
>that is actually readable
What isn't readable to you?

>>9460915
Inconsise notation, symbols not explained properly, no red threat. It is probably a nice book for the guys who participated in writing it, but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic. I discussed this with some of my colleagues some time ago and that was the general consensus. There are new efforts however, at least for quantum mechanics there are now some nice introductory texts for mathematicians.

>>9460954
>Inconsise notation, symbols not explained properly, no red threat. It is probably a nice book for the guys who participated in writing it, but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic. I discussed this with some of my colleagues some time ago and that was the general consensus. There are new efforts however, at least for quantum mechanics there are now some nice introductory texts for mathematicians.
Ah, I thought you meant something on the actual image wasn't readable.

>>9460960
Translation error, sorry.

>>9460954
>but expecting any, lets say, math postdoc to understand the physical concepts and the math stuff he's not familiar with is not realistic
Not really. What physical concepts are you not familiar with exactly?

wtf is a ideal of a polynomial ring?

>>9460976
Do you know what an ideal is?

>>9460954
What's "red threat"?

Are there any other math books for gifted amateurs?

>>9460987
The God Delusion.

>>9460987
Godel, Escher and Bach.

>>9460990
>>9460995
>>>/trash/

>>9460970
A big problem for mathematicians in general is how to interpret the math in the sense what it might mean physically. For example, I don't get the concept of "quantizing". Not that I don't get the math (at least in the QM setting, AQFT is a different matter), but I don't understand the point where people say "yeah, that's a nice quantization". Not sure if I'm getting my point across, fuck me I need sleep soon. On a side note, there are many new updates on QFT stuff on the nlab lately, might be worth lurking there.
>>9460982
Fuck me again, I think it's called "golden thread" in english.

>>9461012
(Pre-)Quantization just ultimately a replacement of the Poisson bracket on [math]C^\infty[/math] functions with the Lie bracket on the Lie algebra of operators on some Hilbert space. This promotes your classical observables to Hermitian operators the eigenvalues of which are the actual measurable quantities.
This amounts to finding conditions on your symplectic manifold [math](M,\omega)[/math] on which you can centrally extend the Lie algebra of [math]C^\infty[/math] functions to that of automorphisms of the Hermitian line bundle [math]B\rightarrow M[/math] on [math]M[/math]. Conventionally this is done with the integrality condition [math]\omega \in H^2(M,\mathbb{Z})[/math] where the curvature of the connection on [math]B[/math] is [math]\frac{1}{\hbar}\omega[/math] but Kostant's construction gives a more general form of this procedure.
A sense in which a quantization scheme is "nice" may be the fact that for free fields you can decompose field operators into creation/annihilation operators (generators of the Heisenberg algebra) and everything you've learned from basic QM falls through, and you can construct S-matrix elements as usual (modulo some more axioms you have to assume such as asymptotic completeness of your Hilbert space). Of course this can't always be done and this depends on whether if Kostant's construction gives you a representation of Heisenberg algebra or not.

>>9461050
Thanks for the explanation! Seems like I should study some more geometric quantization.
Also, the theore behind scattering-""""matrices"""" in aQFT gives me headaches (both the math and the physics part), is it really that well understood in the physics community?

>>9460982

>>9461105
>is it really that well understood in the physics community?
The short answer is no. The long answer is because we haven't completely understood what constitutes an appropriate framework for rigorous QFT yet, which is why we get things like Haag's theorem and regularization problems showing up when dealing with ill-defined S-matrices.
Geometric quantization and AQFT are ways to construct rigorous frameworks for QFT. Though AQFT is more like a "safe than sorry" approach to QFT; it's nice and all but it can't deal with anything other than free fields. If I were to put money on which approach would lead to a better understanding of the formalism of QFT I'd probably go with geometric quantization, though that is not to say that AQFT isn't interesting in its own right.

I understand why the conjuction is true if an only if the propositions are both true, and I also understand how disjunction works

But why is that the conditional is always true when the premise is false? I understand is defined to be that way, but what is the motivation to do so? It feels random

>>9461146
It is random. Just memorize the truth table and move on.

>>9461130
But isn't the S-matrix in (perturbative) aQFT specifically designed to deal with interacting fields? Maybe I misunderstood its article here
https://ncatlab.org/nlab/show/S-matrix
and to be fair I haven't found any other source that I understood even in the slightest. I found a relatively new book though, see pic, might be worth checking out.

>>9461227
>perturbative
>algebraic
I've never seen that desu. My field is TQFT and CFT so I might be wrong on some recent developments in AQFT so can't help you much there.

>>9461255
>>9461255
If I understand the rumors correctly, perturbative aQFT has been under the radar of most people for quite some time, but the interest is rapidly increasing (at least this is true for the mathematicians, not sure if physicists like that kind of approach). Afaik CFT has been and still is a hot topic, some pretty powerfull guys in my department work in that field. But in particular in CFT-talks I never understood the connection to quantum stuff. The math alone was nice though, but to be honest I also don't understand the connections of my field of research (special Kähler stuff & affine diffgeo) to physics, and apparently the people who do should hurry and write some sort of guide cause they are all getting old.

>>9461291
>perturbative aQFT has been under the radar of most people for quite some time, but the interest is rapidly increasing (at least this is true for the mathematicians, not sure if physicists like that kind of approach)
I don't think most physicists are even aware of regular AQFT, though this definitely seem interesting. What category in arxiv would perturbative AQFT go under?
>special Kähler stuff & affine diffgeo
Loop spaces are important in geometric quantization as well as CFT (they facilitate [math]U(1)[/math] gauge invariance), and nice Kähler structures can be put on them to study their geometric properties. And I believe affine diff geo could be used to study connections on moduli spaces which can be used to investigate the existence of wavefunctions and the like (e.g. existence of projectively flat Hitching connection on Verma modules [math]\Rightarrow[/math] existence of conformal blocks).

>>9461324
>Hitching
Hitchin*

>>9461146
>conjuction ... disjunction

Just say logical AND and OR ffs

>But why is that the conditional is always true when the premise is false?

Because conditionals are promises. If I promise my kids pizza if they win the ball game; do I break it if they don't win and don't get pizza; do I break it if they don't win and I buy them pity pizza? No. They only way to break the promise is if they win and I don't buy them pizza.

>I understand is defined to be that way, but what is the motivation to do so? It feels random

Because causation is a bitch and establishing it even more so.

>>9461291
Kahler geometry is important to physics via string theory.

>>9461324
>arxiv
Primary math-ph, secondary hep-th. Sometimes also secondary gr-qc or math.OA. Found what looks like an overview here
https://arxiv.org/abs/1208.1428

For the affine diffgeo stuff, I'll look into your suggestions. What I know is that the Kähler cone of a given Kähler manifold, e.g. some Calabi-Yau mfd., carries the structure of a Riemannian centro-affine manifold, which I guess connects it somehow to certain moduli-spaces in supergravity. But I never really understood the step from some 10d spacetime (where 6d's are a CY mfd.) with some action to the corresponding sigma model. I think that this might be the key thing in linking the part of affine diffgeo that I study, CY- resp. Kähler-geometry, and the physical interpretation. Will report back once I can explain it, might take longer than this site might live though...

>>9461371
>But I never really understood the step from some 10d spacetime (where 6d's are a CY mfd.) with some action to the corresponding sigma model.

Read chapter 3 of this book.
http://www.claymath.org/library/monographs/cmim04.pdf

>>9461371
>https://arxiv.org/abs/1208.1428
Thanks anon. I'll try to read this after reading/understanding the Ayala paper (i.e. never).

>>9461378
Thank you, I'll try my best!
>>9461390
I don't know which paper you are referring to, but google points to cats. Honestly, pure category theory is even more confusing than physics. Stay strong m8, here's a bunch of arrows in case diagrams don't commute [math]\rightarrow \rightarrow \rightarrow \rightarrow \rightarrow[/math]

>>9461432
>I don't know which paper you are referring to
The one on the cobordism hypothesis: https://arxiv.org/abs/1705.02240..
Ayala's paper talks about a necessary condition (i.e. the existence of factorization homology functor with adjoints) for constructing partition functions of general TQFTs with one (or at least finitely many) generating object(s) in the category of cobordisms. This could in principle point to a concrete construction of a quantum invariant needed for 4D TQFTs and may hopefully circumvent the problem of positivity for 4D UTQFTs, which is a major obstacle for AdS/CFT.

What is it about physics that makes it so tractable?

what did terrence tao prove about newman's number? is it a real proof and if so why is it not discussed more on serious boards like physicsforums stackexchange etc?

>>9461482
>https://arxiv.org/abs/1705.02240
I really admire people who got a feeling for that kind of algebraic thinking. If you are able to understand that kind stuff, then that aQFT-book will probably be a piece of cake for you.

>>9461482
>functor

>>9461536
>what did terrence tao prove about newman's number?
It's non-negative.

> is it a real proof and if so why is it not discussed more on serious boards like physicsforums stackexchange etc?
https://arxiv.org/abs/1801.05914

>>9461537
>If you are able to understand that kind stuff
Lmao you flatter me.

>>9459925
Ahlfors complex analysis

I'm trying to work on some combinatorics.

I have n objects, some recur multiple times.

I want to find the number of ways I can make non-unique r-length sequences.

Am I correct in saying that this answer is just (n choose r), using the Vandermonde identity in its general form?

>>9459073
Which is the best non-dry yet rigorous book for self teaching Measure Theory?

How does that drawing make sense? Yao has his own Fields Medal, Perleman didn't accept his, and Yao never said Perlman didn't deserve one.

>>9461868
Yau, sorry.

>>9459928
baby rudin is a really excellent book. maybe it's hard if you don't know any analysis but it's really fantastic

what are some good books for getting into p-adic numbers and analysis over [math] \mathbb{Q}_p [/math]?

>>9460785
what kind of department are you in where people don't know peano axioms

>>9461898
gouvea
koblitz

>>9459278
>>9459308
Why would I want to work with animals?

In the end, this turned out to be shopped. But what's the solution?

>For all natural numbers k, show the inequality holds [eqn]\frac{1}{2(k+1)} < \int_0^1 \frac{1-x}{k+x} dx < \frac{1}{2k}[/eqn]

Is there anything interesting to mention regarding (connected) topological spaces (non-trivial sets are either open or closed) and classical logic (non-trivial statements being either true or false)?

If so does this carry over to any analogous spaces for other logics (say three valued, or real valued)?

>>9462466
>logic
not science or math

What are the point of Lie groups and algebras? I'm taking a course in Riemannian geometry and I really don't see where this is going

>log in to /mg/
>no (you)'s
>log out

>>9462466
i dont know anything about it but ive heard some people say logicians are some of the few people who still care about point set topology as such

>>9462466
you might want to look at topos theory

>>9462466
Topological spaces are models for intuitionistic logic, as in toposes.

>>9462501
Groups of rotations and other physical operations have differential structure. They come up all over the place in physics.

>>9462543
I don't care about physics, i meant what are the use of them in mathematics.

>>9462538
>>9462530
Thank you, friends

>>9462563
Geometry, obviously. If you don't care about geometry then you won't have much use for Lie groups.

>>9460771
As an immediate corollary, yes

>>9459461
>you will never spend your Phd trying to define the category of axioms

Can anybody tell me where to find a proof for the fact that for any isomorphism, [math]\phi[\math], between finite Symmetric Groups [math]S_x[/math] and [math]S_{x'}[/math], the isomorphism is of the form [math[\phi(\sigma)=\tau\sigma \tau^}-1})[\math]for some [math]t \in S_x[math]?

Trying >>9462595 again,

Can anybody tell me where to find a proof for the fact that for any isomorphism, [math]\phi[/math], between finite Symmetric Groups [math]S_x[/math] and [math]S_{x'}[/math], the isomorphism is of the form [math]\phi(\sigma)=\tau\sigma \tau^}-1})[/math]for some [math]t \in S_x[/math]?

>>9462595
An isomorphism between finite symmetric groups has to be an automorphism.

>>9462595
You won't find such a proof since the statement is false.

>>9462601
Where is the mistake? Is there a similar statement that I might have mistaken this for?

>>9462612
>Where is the mistake?
There exists an outer automorphism of S_6.

>Is there a similar statement that I might have mistaken this for?
It's true for n \neq 6.

>>9462600
This statement is meaningless.

>>9462636
>This statement is meaningless.
How so?

>>9462637
Well, maybe I should say it's just false.

An automorphism is a map from an object to itself. Isomorphic objects can be identified (in which sense what you said is trivially true) but the category of groups has non-equal isomorphic objects as typically defined (take S({1}) and S({0}).

But talking about equality of objects in a category is essentially meaningless. What you mean to say is that automorphisms of symmetric groups are inner (for S_n, n != 6).

>>9462641
>category of groups
This statement is meaningless.

>>9462642
This guy seems to be a faggot.

>>9462647
>This guy seems to be a faggot.
I'm not a "guy".

>>9462647
>This guy seems to be a faggot.
Why the homophobia?

Has anyone computed the cohomology of the category of axioms? I can't find it in the literature

>>9462612
https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#The_exceptional_outer_automorphism_of_S6

>>9462796
It's a trivial use of the Gödel spectral sequence, I doubt anyone would even bother to write it down.

>>9462568
yes i know they're used in geometry you fucking retard, I stated i was learning about them in my fucking geometry class. My question is what are their use in geometry??

>>9462501
Topology

>>9460987
Weinberg, the quantum theory of fields
hartshorne, algebraic geometry

>>9460813
Constructive mathematics is a generalization of classical mathematics.

>>9461917
Animals are far more intelligent than humans.

>>9460747
The dual of a vector space is the set of linear functionals V->K. The dual basis is a set of dual vectors so any other linear functional can be written as a linear combination of it, and that it gives 1 if used on its related unit vector and 0 to all other unit vectors.

>>9462867
Literally the same thing as I said before, they are transformations of space. Anytime you have a space you have a Lie group.

How should I intuitively think of the torsion part of a homology group? I understand that the rank of the torsion-free part is something like the number of [math]n[/math]-dimensional holes. Is there a similar interpretation for the torsion part? Does it have to do with orientability? For example, how do I interpret the fact that the Klein bottle has [math]H_1(X) = \mathbb Z \oplus \mathbb Z / (2) [/math]

>>9463394
>and that it gives 1 if used on its related unit vector
Disgusting.

>>9463911
>Disgusting.

>>9463394
>The dual basis
This doesn't necessarily exist.

>>9464120
>>Disgusting.
It is disgusting, in fact it's incorrect (not every vector space admits a norm).

>>9461868
Yeah, I don't know if I should trust that article. It seems absurdly biased to include that image.

>>9464132
Then just outright post that he should have said "basis element" instead of "unit vector". It's not that hard to communicate, you know.

>>9464203
>Then just outright post that he should have said "basis element" instead of "unit vector".
Why would I tell him/her that he/she should have said an equally incorrect statement?

>>9463906
As the name torsion suggests, the torsion part suggests how twisted the boundary mapping is, and you can see it in the fact that, say, when mapping an n-cell's boundary, the torsion group in the homology is typically the degree of the map

>>9460980
Yes, I don't know what a polynomial is.

Someone help a brainlet out. I'm not seeing this one.

>>9464624
[math]\frac{n!}{(k+1)!(n-(k+1)!)}=\frac{n!}{(k+1)k!((n-k)-1)!}=\frac{n!(n-k)}{(k+1)k!((n-k)-1)!(n-k)}=\frac{(n-k)n!}{(k+1)k!(n-k)!}=\frac{n-k}{k+1}\cdot\frac{n!}{k!(n-k)!}[/math]

>>9464624
nvm, I got it.
>Break your head on this for 2 hours
>Post it
>understand it just a minute later.
ffs.

>>9464646
>LelTeX
I leave /sci/ for a year and it still shits itself if you don't put spaces at certain points. Hiroshimoot do your job already.

[eqn]\frac{n!}{(k+1)!(n-(k+1)!)}= \frac{n!}{(k+1)k!((n-k)-1)!}= \frac{n!(n-k)}{(k+1)k!((n-k)-1)!(n-k)}= \frac{(n-k)n!}{(k+1)k!(n-k)!}= \frac{n-k}{k+1} \cdot \frac{n!}{k!(n-k)!}[/eqn]

Is there any notion of taking the n-th dual of a vector space? I've learned about the dual of a dual: [math](V^{*})^{*}[/math], but is there anything special about the triple dual, quad dual, ... etc?

Why couldn't you generalize the notion of a limit to include numerical PDE solutions, letting you prove the existence of Navier Stokes solutions?

>>9461548
Yukari Math Friend, how do I get into sympletic geometry! I know differential geometry, commutative geometry, and topology right now! I still need to learn more algebraic geometry

>>9465898
In a finite dimensional vector space, I don't believe there is anything that special about the n-th dual, as I think they are all isomorphic to each other. I think weird stuff starts happening with infinite dimensional vector spaces.

>>9461948
Squeeze theorem?

>>9461948
yeah i think the squeeze theorem proves the exterios ones and you can integrate and prove either identity right?

>>9466049
hm, yes, I see. Would you happen to be versed in infinite dimensional vector spaces? I've only ever studied finite one.

>>9465898
the even duals are isomorphic to V, the odd duals are isomorphic to V*

>>9465980
The notion of a limit is already pretty darn general as it is, and the problem with NS isn't numerical solutions but blow ups

>>9466023
Study classical mechanics.

>>9466239
What book!

>>9466262
Symplectic Techniques in Physics by Guillemin and Sternberg.

>>9466277
Will I be fine, I dont know any physics

>>9466282
The first part of the book will teach you extremely basic physics from optics to classical mechanics.

>>9466292
Thanks friend! Do you have steam or discord, lets be buddies!

>>9466307
Nope, sorry.

>>9466120
*naturally isomorphic

>>9459073
I get that mathematicians are all glory hogs, but wasn't Yau one of the guys who nominated Perelman for that medal in the first place?

>>9466556
But Perelman wasn't a glory hog.

I don't get the difference between "normal" and "strong" induction.

Is the only difference between then saying
"If it's true for all <n, then it's true for n"
and
"If it's true for n, then it's true for n+1"

>>9467081
Yes, pretty much. However, you can prove that these two are equivalent.

why are proofs about properties of markov chains so fucking tedious

define "mathematics"

>>9467166
Because your textbook is not using enough linear algebra, probably.

>>9459925
Ahlfors - Complex Analysis
and Logic, if there is a english translation available of
"Ebbinghaus - Einführung in die mathematische Logik"
That dude also has an EXCELLENT book on set theory "Einführung in die Mengenlehre"

>>9460785
Wow, glad we are on different departments I guess.
I am always surprised how even the guys who work in (sub-)fields not even touching diffgeo or logic in a meaningful way still have the education about logic, GTR and differentialfroms in such a precise and deep level that blows my mind every time.
Better be an educated autist than an undeucated but down looking PhD (which the department gave you after 2 years cause you ain't no good and they want somebody better to pay this position for).

>>9459811
https://terrytao.wordpress.com/2018/01/27/polymath15-first-thread-computing-h_t-asymptotics-and-dynamics-of-zeroes/

>Polymath15, first thread: computing H_t, asymptotics, and dynamics of zeroes

>This is the first official “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant [math]\Lambda[/math]. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

>The proposal naturally splits into at least three separate (but loosely related) topics:

>Numerical computation of the entire functions [math]H_t(z)[/math], with the ultimate aim of establishing zero-free regions of the form [math]\{ x+iy: 0 \leq x \leq T, y \geq \varepsilon \}[/math] for various [math]T, \varepsilon > 0[/math].
>Improved understanding of the dynamics of the zeroes [math]z_j(t)[/math] of [math]H_t[/math].
>Establishing the zero-free nature of [math]H_t(x+iy)[/math] when [math]y \geq \varepsilon > 0[/math] and [math]x[/math] is sufficiently large depending on [math]x[/math] and [math]\varepsilon[/math].

>tfw you realize that getting good at math is really fucking hard and you have no chance of ever catching up to someone who started early and worked properly
>tfw you give up and just start working on numerical solutions to pdes with the cs dept

oh well

Given an if and only if type proof, if you mess up the forward but get the backward, should it at least be awarded 0.5/2 or 1/2 marks?

>>9468375
Why 0.5/2?

>>9468375
You are told to prove an iff statement, and you prove an if statement. Therefore you don't do what you are told to do, and so you would get 0 if I was grading you.

>>9468375
You should be awarded 0 marks since you didn't prove it. If you rewrite the statement as an "if" proof then you get full marks for that proof, but that's not what you were told to do.

>>9468375
It depends on how harder one direction is from the other.
If one direction of the proof is literally one line starting with "Obviously", and that's the one you proved, then you'll get ~0 points.
If both directions are equally hard, then you should get hald the points.

>>9468387
>>9468389
Now, that's what I call autism.

>>9468403
Those posts are correct though, you didn't prove the statement you needed to prove.

>>9468403
No. It is quality control.

>>9468410
>>9468412
dumb

>>9468530
Whatever you say. Some people just aren't physically capable of understanding.

>>9468412
This is the first time I've been on this board in years and years. This was on the front page.


You guys are alright.

>>9468657
I'm not a "guy".

>>9468660
Do your chromosomes corroborate your statement?

>>9468666
I really, really need to check this place out more often.

>>9468657
And I'm not a "you".

>>9459073
Fuckin yau lol. They did him dirty in this piece. He's a legitimate mathematician but the article made him seem like he steals all of his work.

>>9468954
he was a little bitch for wanting credit. but yaus is an animal. apparently he took 7 classes while auditing others when he was getting his phd at berkeley. i can barely handle two

>>9468984
why is someone who wants rightfully deserved credit quote a little bitch unquote?

>>9469008
>rightfully

>>9469008
>rightfully

>>9469008
>rightfully

>>9469008
>rightfully

>>9469008
>rightfully

>>9469008
>rightfully

>>9469008
>rightfully

That is such a stupid drawing and should have never been published. It makes me doubt the entire article.

>>9469011
>>9469066
>>9469199
impressive

>>9469008
Chinese shill detected

>>9469008
Fuck off Yau

https://arxiv.org/pdf/1407.4897v3.pdf

What are the chances that twin primes is false and the "near miss" scenario occurs? (Thm 1.5)

redpill me on mirror symmetry

>tfw you realize that setoids are all we need and all the hand-wringing about quotient types and higher inductive types is just category theory autism

thank you based bishop for solving all foundational problems 50 years ago

>>9470070
read
https://ocw.mit.edu/courses/mathematics/18-969-topics-in-geometry-mirror-symmetry-spring-2009/lecture-notes/

>>9459073
wtf is a manifold, can someone explain it to me properly?

>>9470697
a thingy that looks like R^n locally

>>9470697
A second-countable Hausdorff space locally isomorphic to R^n.

>>9468387
with this logic you would have to mark every possible question with full points or zero points, never in between

>>9470757
>with this logic you would have to mark every possible question with full points or zero points, never in between
Correctness is binary.

>>9470768
>Correctness is binary.
so ?

>>9470757
And this would be problematic because of what? Of course, some exceptions could be made if one direction is superduper much harder than the other, but then it would be quite surprising if one could only prove the hard direction. But, assuming for example that zâ ran out of time doing the hard direction, then I could see myself giving zõr some points, if the other problems were correct.

>>9470634
>foundational problems
No such thing.

>>9471073
Not anymore :^)

>>9470952
>zâ

>>9471080
>:^)
Redditors are not welcome here. Proceed to >>>/r/eddit/

Hey guys, Im a physics + math major. I'll have finished with enough courses to have both a pure and applied math degree alongside my physics bs degree. So Ill have the prerequisites for either applied or pure math masters. Would anyone like to make a recommendation/insight as to which I should pursue?

Heres a look at the differences

Applied
>advanced real analysis I
>advanced numerical methods I, II
>advanced applied math methods I, II
>advanced PDEs I,II
>optimization
>advanced functional analysis I
>advanced applied statistics


Pure
>advanced linear algebra
>theory of numbers
>advanced abstract algebra I,II
>advanced real analysis I,II
>advanced functional analysis I,II
>advanced graph theory
>theoretical statistics
>advanced probability


Actually, after typing that out I know which Im leaning toward but Ill post this anyway if anyone has any comments

>>9471118
>applied math
No such thing.
>advanced
Seems like garbage if you ask me. Most of these topics are "advanced" enough without needing to explicitly mention it. While the others simply can't be "advanced" since they don't exist, like "advanced applied math methods" for example.

>>9471118
Could you say what will be teached in each class?

>>9471118
I feel like your university's course titles don't actually have the word "advanced" in them and you just put it in all of them to make yourself look intelegint.

>>9471079
People should respect each other's pronouns. I am not here to trigger anyone, so I use neutral pronouns. If you have a problem with that, I suggest you practice some introspection.

>>9471186
Ps your post made it hard for me to breath.

>>9470634
>the category of abelian groups defined in type theory without quotient types need not be an abelian category
That's properly autistic.

How to prove if U is a subspace of V then dim(V) = dim(U) (finite dim) implies U=V .

something to do with extension of basis in U perhaps? me brainlet, me don't know

>>9471225
don't define everything w.r.t """actual""" equality and you'll be fine

>>9471259
Index the basis elements and map each basis element of V to U so that ith basis element of V is sent to the ith basis element of U. Then you prove this defines a linear map, and you construct an inverse in the obvious way.

>>9471259
The basis B for U is linearly independent with n=dim(U)=dim(V) elements. Isn't this enough?

>>9471284
I had this particular brainwave, but it didn't seem like enough.

>>9471288
B is clearly linearly independent, as it is madw by elements in V such that, for no scalars k1,...,kn (not all zero),

k1b1+k2b2+...+knbn=0

As it is linearly independent and has n distinct elements, it must be a basis.

>>9471265
That is how it's usually done where quotient types don't exist. The category of abelian groups still fails to be abelian. It's pretty obvious that setoids are an inferior notion.

>>9471297
how precisely does this show U=V however?

>>9471358
Because U and V have equal basis, their elements are the same. Try to think on some example, like in R^n.

>>9471118
lmao how does applied not even have linear algebra

take:
>linear algebra
>abstract algebra I,II
>real analysis I,II
>functional analysis I,II
>theory of numbers
>maybe dabble in some PDEs

>>9471259
Remember that mathematicians abuse notation and [math]U=V[/math] usually means [math]U\cong V[/math], ie that they are isomorphic. In this particular case you can construct an isomorphism that sends the basis vectors to the basis vectors bijectively, and this will turn out to be linear and a bijection of the spaces.

>>9459925
Why Munkres instead of Hatcher anon?

If I have three projective points in a plane that aren't colinear can I assume without loss of generality that they are unique? Otherwise if one is the other, a unique line can be formed between that and the remaining, causing colinearity.

>>9471430
>Remember that mathematicians abuse notation and U=V usually means U≅V
It's actually an equality in this case.

>>9471588
any takers? (please)

>>9459073
I am taking a topics course focused around this correspondance, https://ncatlab.org/nlab/show/Donaldson-Uhlenbeck-Yau+theorem

Is anyone familiar with it?

>>9471707
the span of those three vectors form a subspace of V for any dimension greater than or equal to two, r-right?

>>9471125
Show me on the doll where your applied math professor touched you

>>9471739
How does something non-existent touch an existing object?

>>9471259
All modules over a field are free, therefore projective. Consider the short exact sequence [math]0\rightarrow U\rightarrow V\rightarrow V/U \rightarrow 0[/math]. Dimension is an additive invariant, so [math] V\cong U\oplus V/U[/math] gives the dimension of [math] V[/math] is the dimension of [math] U[/math] plus the dimension of [math] V/U[/math]. But there's only one vector space of dimension 0, so [math] V/U=0[/math]. So [math]U=V\oplus 0=V[/math].

>>9471259
Let [math]\iota: U \hookrightarrow V[/math] be the canonical inclusion, and let [math]M,N[/math] be [math]3[/math]-manifolds such that [math]T(M)
= U[/math] and [math]T(N) = V[/math] by the TQFT functor [math]T: (\mathscr{B},\mathscr{A})
\rightarrow \mathscr{M}[/math]. Suppose [math](W,f:M \rightarrow N)[/math] is the cobordism between [math]M[/math] and [math]N[/math] such that [math]T(W) = \iota[/math]. Denote by [math](v_X, X \rightarrow \emptyset) [/math] the cobordism such that [math]T(v_X)[/math] is the inner product by the vector [math]v_X \in T(X)[/math], and let [math]v_X^*[/math] be its dual.
By composition the cobordism [math](v_M^* W v_N, \emptyset \rightarrow \emptyset)[/math] is sent to the map of multiplication by the scalar [math]\langle v_M, \iota(v_N)\rangle \in R[/math]. Suppose [math]\operatorname{dim}V = \operatorname{dim}U[/math], then the multiplication [math]\langle v_M, \iota(v_N)\rangle: E \rightarrow E[/math] has non-zero rank for any vector space [math]E[/math]. However this is only possible if [math]\langle v_M,\iota (v_N)\rangle = 1[/math], hence the cobordism [math](v_M^* W v_N, \emptyset \rightarrow \emptyset)[/math] is the identity and hence by the functoriality of [math]T[/math], [math]\iota = \operatorname{id}[/math].

>>9471758
>All modules over a field are free
That's not true.

>>9471805
yes it is

>>9471824
>yes it is
No it's not.

>>9471827
yes it is

>>9471791
dumb and inelegant. kys

>>9471837
>inelegant
Wrong.

>>9471843
Then prove it using your mathematical pants/cobordisms you autistic quantum weeaboo.

>>9471876
Trivial and left as an exercise.

>>9470634
>no path interpretation

>>9471669
I know, but it is easier to think about it in terms of isomorphisms

>>9471462
you cant do algebraic topology without general topology

>>9471707
bump

Is there any notion of defining arithmetic on cardinals for hyper-4 operations and beyond? Tetration, etc.

Say you have a bunch of equally spaced points [math] x_0,x_1,\ldots,x_n [/math] (increasing order) and you want to calculate the (unique) interpolating polynomial P of degree n, so that you evaluate that polynomial at some x.

Why is "Newton's Forward divided difference formula" considered more appropriate when x is close to x0,
while
"Newton Backward divided difference formula" is considered more appropriate when x is close to xn
?

Can somebody explain what a polynomial is? How does one "evaluate" an element of a free commutative algebra?

>>9472628
polynomials are a spook

>>9472630
As in "intelligence agency employee"? Please don't toy with me.

>>9472628
a polynomial is a "sentence" in the language of rings

>>9472650
Is there a non-logical interpretation of them?

>>9472628
>what is the evaluation homomorphism [math]\phi_a: F[x] \rightarrow F[/math] where [math]f(x)\mapsto f(a)[/math] for any [math]a\in F[/math]

>>9472659
>Is there a non-logical interpretation of them?
What do you mean?

>>9472663
>>what is the evaluation homomorphism ϕa:F[x]→F where f(x)↦f(a) for any a∈F
Who are you quoting?

>>9472667
*Whom

>>9472628
You can consider it as a sequence (the elements being the coefficients of the polynomial) such that it has a finite amount of non-zero elements and you define addition and multiplication appropriately, but that's just autistic.
Just consider them as formulas that are waiting to be evaluated.

>>9472667
I'm not a "you".

>>9472663
What do you mean precisely by [math]f(x)\mapsto f(a)[/math]? Is this not a constant map?
>>9472678
>but that's just autistic
Why?
>Just consider them as formulas that are waiting to be evaluated.
I'm not satisfied with an engineer-like explanation of them.

>>9472684
Yes it maps to a field element i.e. "constants" if you want yo use engineer-like nomenclature.

>>9472681
>not
excuse me?

>>9472686
>Yes it maps to a field element
So what is [math]f[/math] precisely?
>"constants"
I didn't use this word in my posts. I don't even know what it means. I said "constant map".

>>9472681
>I'm not a "you".
What are?

>>9472695
Sentence requires a subject sweetie, what you just wrote is ungrammatical.

>>9472697
What you wrote is just unphysical abstract garbage.

>>9472702
nice adverb there bud, want to try again?

>>9472684
>>but that's just autistic
>Why?
Because it only obfuscates what a polynomial actually is and why we study them.
It serves no purpose.

>I'm not satisfied with an engineer-like explanation of them.
I gave you a "mathematical" definition of them. Here for more details:
https://math.stackexchange.com/a/216480

>>9472628
the polynomial ring over a commutative ring R is precisely the symmetric tensor algebra over R^* (the dual space of R).

>>9472758
>is precisely
What's the difference of being something and being precisely something?

>>9472763
>differentiable functions are continuous functions
>differentiable functions are precisely continuous functions
only one of these claims is true

>>9472326
>no whiz-foddles and smebble-crumpets

>>9472435
wildberger does

>>9472717
>what a polynomial actually is
Can you explain what it is?
>It serves no purpose.
How do you show that they exist then?

childhood is obsessing over exact symbolic solutions to integrals
adulthood is realizing that numerical computation is far more important

>>9472598
bump

>>9472628
A polynomial in n-variables is a regular function on affine n-space.

How would it look to grad schools if I haven't taken Algebra or Topology, but have taken Algebraic Topology and gotten an A?

>>9473584
I would be pretty much as concerned about your shaky foundations as if you hadn't taken it.

Passing algebraic topology does not mean that you understand algebra or general topology except in the very limited forms required for your course. You could survive an intro algebraic topology course not even knowing what a ring is, let alone anything about them.

>>9473625
An algebraic topology course should include cohomology, and one of the most important reasons for studying cohomology as opposed to homology is its ring structure.

>>9473584
>How would it look to grad schools if I haven't taken Algebra or Topology, but have taken Algebraic Topology and gotten an A?
It depends what you're applying to grad school for. Having A's is standard but if it's grad school for mathematics then they certainly won't care about algebraic topology.

>>9473633
>grad school for mathematics then they certainly won't care about algebraic topology.

Why would grad schools for math not care that I took and did well in a graduate math course?

>>9473637
>Why would grad schools for math not care that I took and did well in a graduate math course?
What do you mean? Of course they would.

>>9473638
>they certainly won't care

>>9473640
Right, about algebraic topology.

>>9473643
which is a graduate math course

>>9473645
>which is a graduate math course
No it's not.

>>9473647
yet it is

>>9473649
>yet it is
Wrong.

>>9473653
wonder what im enrolled in then

>>9473655
>wonder what im enrolled in then
Something outside of mathematics, it seems.

>>9473657
Weird, the course says "MATH"

>>9473659
>Weird, the course says "MATH"
Someone in your department must have made an error.

>>9473665
Maybe I should inform the algebraic topologist teaching the class that his course isn't math.

>>9473670
>Maybe I should inform the algebraic topologist teaching the class that his course isn't math.
I'm sure he/she is well aware.

>>9473672
Idk, he is an asst. professor in the math department.

>>9473670
And he'll respond "I'm not a 'his'."

>>9473679
>Idk, he is an asst. professor in the math department.
It's not too uncommon for professors to teach outside their department.

One of my math professors also taught in the philosophy department.

>>9473680
Pretty sure he is too old to be aware there is any other option.

What is the best way for a brainlet to learn trig? I've got a test on Monday and I want to be prepared. I've been studying but the textbook we use for the class is trash, anyone have any recommendations for something that expains in better, but also isn't too long since I only have a few days? Things like solving triangles using the Law of Cosine and the Law of Sines I feel okay with. Rads make sense, no problems there. What is mainly giving me problems is the Unit Circle. I don't really understand it.

>>9474229
For trig you need these few things:
Sohcahtoa
A trigonometric circle (one cadran should be enough)
And maybe to go further Euler's formula for sin and cos, and their Taylor series

>>9473645
>graduate math course
What sort of school for brainlets teaches something you should have learned in high school in a "grad" "course"?

>>9474229
>Law of Cosine and the Law of Sines
>the Unit Circle
These are currently research topics, I don't think you will find answers here.

Lads I have a PhD interview next week at Oxford, any tips?

>>9474229
>Rads make sense, no problems there. What is mainly giving me problems is the Unit Circle. I don't really understand it.
How is that even possible?
What does "Rad" mean in your mind?

>>9473647
>implying Algebraic Topology is not advanced enough for a graduate course
What is this heresy?

>>9474942
>>implying Algebraic Topology is not advanced enough for a graduate course
I didn't imply that.

What exactly are "generalized coordinates" in mathematics? I highly subspect it has to do with differential geometry as something something a parametrisation of a submanifold of R^n.

>>9475013
>What exactly are "generalized coordinates" in mathematics?
https://en.wikipedia.org/wiki/Generalized_coordinates

>>9475028
Lol nigger I've seen that fucking definition a thousans times in physics texts, but I'm asking if they are just a subset of a more well defined and formal concept in differential geometry.

>>9475036
>Lol nigger
Why the racism?

>>9475037
>Why the racism?
Why the memes?

>>9474942
Most "graduate" courses (algebraic topology included) belong in high school.

>>9469008
>leftfully

>>9475028
this article reads like physicist pseudo-formal nonsense. If there is a rigorous definition I don't see it. "Degrees of freedom" usually means the dimension of some manifold or moduli space.

>>9475616
read the formulas then and ignore the blabber????????

>>9475013
When you have a manifold and select a specific coordinate chart, you have coordinates on your manifold. You can then change your coordinates if you have a different chart via transition mappings.

is it true that Perelman stole from chinese mathematicians?

>>9476097
>manifold
Is this well-defined though?

>>9476097
So why physicists avoid talkinng aboit them in this way? At least mention it. Differential geometry and forms shoukd be a must fir physicists.

>>9476129
yep

>>9476134
Why?

>>9476132
because they would need to take a huge detour to explain most of it. I don't do physics and I think it's dumb not to do it that way but that's the way it is.

There's this interesting lecture series by a brilliant lecturer that starts from the literal beginning and builds all the way up the differential geometry used in physics: https://www.youtube.com/watch?v=V49i_LM8B0E&t=4s
Highly recommended watch (It's bearable at x2 speed)

>>9476135
because the reals are well-defined

ok so I've got four chemicals to make a batch of 21 pills I'd need
4.2g l theanine
4.2g caffeine
1g gluteamine
6.3g alpha GPC

now the costs for each items are as follows
l theanine 50g/10bong
caffeine 100g/8bong
glutamine 500g/9bong
alphaGPC 25g/17 bong

I'm trying to work out what one pill costs me to make and please explain the maths as simply as you can

Explain to me Hilbert's Nullstellensatz like I'm a dumbass

>>9476140
Thanks bro.

>>9476167
gunna take an uneducated guess. Do you put this in a matrix and do the Gauss elimination and row echelon from stuff?

>>9476222
this is my first time on /sci/ lad and I only got c for maths GCSE and now I need maths so yeah

>>9476185
I'll do almost the same thing. I'll explain it to you like I'm a dumbass: when will you ever use it outside school?

>>9476230
I want to use it for school cos i wanna do a phd, so what do i care what it can be used for?
Anyways, the answer is: coding and error correcting codes

>>9459278
>working with brainlets

I refuse.

>>9459278
Go away Yukarifag

>>9476234
It gives you an order reversing correspondence between ideals and algebraic sets. Find out what those are and then you are done.

>>9476019
It says

>It is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context

This makes it sound like you start with an ambient n-dimensional manifold with some coordinate chart, then you take an m-dimensional submanifold defined by some equations, and you find a coordinate chart for it in R^m.

But it also says "However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations."
which means there is essentially nothing special about "generalized" coordinates, they're just coordinates.

I have discovered what a polynomial formally is. Can anyone please tell me how to "evaluate" an element [math]f \in R[X][/math]?

>>9476425
[math]R[X] \times R \to R, (f(X), a) \mapsto f(a)[/math]

>>9476485
>[math]f(x)[/math]
>[math]\in R[x][/math]
Found the retard undergrad.

>>9476425

A polynomial is a word in one variable in the theory of R-algebras.

You evaluate it by choosing a model and an interpretation for the variable.

>>9476485
>[math]f(X)[/math]
>[math]f(a)[/math]
I don't understand what you mean by this.
>>9476512
>>9476518
I'm not well versed in models and their interpretations. And why does it have to be in one variable?

>>9476529
>why does it have to be in one variable

A polynomial doesn't but you asked about [math]R[X][/math] which is polynomials in one variable.

A word is literally just any formal combination of a theory's operations (up to equivalence in that theory).

Interpreting it just means you choose what +, -, and * are, along with plugging in a particular value for X.

>>9476502
>Found the retard undergrad.
Why the ableism?

Taking abstract algebra,probability,and analysis I this semester. Which one is generally considered to be the most time consuming and/or hardest?

>>9476230
dumb moeshitposter

>>9477122
I-I'm not dumb!

>>9477122
>moeshit
This is not well defined.