/mg/ - math general

[Anonymous No.10803629]

Previously: >>10795697

2^69 dimensional tic-tac-toe edition

[Anonymous No.10803644]

Threadly reminder to work with physicists.

[Anonymous No.10803659]

>still want to do that pure math PhD anon?

[Anonymous No.10803672]

>>10803659

Yes, but I know I probably can't

[Anonymous No.10803678]

>>10803629
What is manifold in topology?
I am trying to understand it while using bulletphysics.

[Anonymous No.10803681]

>>10803644
wouldn't a mathematician have an easy time moving to physics, but not vice versa?

[Anonymous No.10803739]

>>10803678
A space that is locally euclidean. A simple example would be the (surface of a) sphere, in particular the Earth. Look around yourself and see the geometry you were taught when you were a kid: lines are straight etc. Then use your connections to get a flight on a U2 or a Blackbird to see the Earth from a higher perspective to notice how the euclideanness you experienced only moments ago doesn't hold in a large enough scale. That is what can happen (but doesn't need to, as the euclidean spaces themselves are manifolds) because the euclideanness condition is local. Also, depending on the author, there may be some separation and countability axioms that need to be satisfied, but your post makes it seem like mentioning their details would be a mutual waste of time.

[Anonymous No.10803760]

>>10803678
If you were an ant living on it anywhere, you wouldn’t be able to tell that it is not flat

[Anonymous No.10803815]

>>10803659
Where did you get this? What was the record of the guy applying?

[Anonymous No.10803886]

>>10803681
A good academic can pick up any subject by sheer interests. Most people are cucks who are afraid of going beyond their niche.

[Anonymous No.10803942]

>>10803815
it's from /r/math, it's still the top post if you want to see yourself.
dude did his PhD is graph theory and this was the result (those are teaching professor positions, incidentally, he apparently has a strong background teaching so applying right out of PhD isn't out of the question)

[Anonymous No.10803960]

>>10803942
>PhD is graph theory
Oh, that explains it.

[Anonymous No.10803971]

>>10803960
>implying your alg geo phd is going to fare better

[Anonymous No.10803984]

>>10803971
There's no need to be so defensive. You can always switch out to engineering and apply your graph theory PHD there.

[Anonymous No.10804000]

>>10803942
>/r/math
Refer to >>>/r/eddit/.

[Anonymous No.10804061]

>>10804000
>Refer to >>>/r/eddit/.

[Anonymous No.10804079]

>>10803942
Read the thread, he genuinely fucked up by only applying to ~70 positions. Many of the younger profs I know applied to FAR more.

[Anonymous No.10804091]

>>10804079
>Read the thread
see >>10804000

[Anonymous No.10804121]

>>10804000
the math subreddit is pretty good and has interesting posts once in a while. because I'm not a retard I don't engage in website tribalism.

[Anonymous No.10804166]

>>10804121
>the math subreddit is pretty good
see >>10804091

[Anonymous No.10804181]

>>10804000
>>10804091
>>10804166
have sex

[Anonymous No.10804200]

>>10804181
Dilate.

[Anonymous No.10804300]

>>10804200
already did today, you?

[Anonymous No.10804308]

>>10804300
with your bf?
>>10804121
the math subreddit is fine at best, many of the posts are just people trying to look smart or putting the wrong thing in the wrong place though. at least we have fun with it here on /mg/.

[Anonymous No.10804327]

>>10803678
a manifold has 3 properties:
- hausdorff: so any two separate points are actually separate, you can draw little neighborhoods around each one which do not intersect
- second countable: so there is a sequence of open sets which can be used to write any open set via unions (think how any real number is a limit of rational numbers, and there are countably many rational numbers)
- locally euclidean: so at every point there is a neighborhood which is homeomorphic to R^n for some n. what this means is that you can cut out a little ball around any point and stretch/twist that ball so that it looks like part of physical n-dimensional space.
any space like this is a topological manifold
once you have this last property, you can define a manifold via an "atlas" of "charts" which is just a collection of functions between parts of your manifold and R^n. the functions should be homeomorphisms and for every point in the manifold there has to be at least one function (chart) which takes care of it. it's just like having flat maps for earth.
if the charts in your atlas are in fact differentiable, or more precisely if the composition of one such map and the inverse of another as a map from R^n to R^n is differentiable, then your manifold is a differentiable manifold. if the charts are smooth, then you get a smooth manifold.
if you attach a metric to you manifold (an inner product in the tangent space at each point) then you get a riemannian manifold.

[xxXSuiseisekiXxx No.10804344]

>>10803681
>moving to physics
There's a whole host of ideas already mined that you have to be aware about, to not waste time.
I'm around 14 years into it. And I'm still finding absurd thoughts to possible physics mechanisms by 'theorists', from ages that just shouldn't have that kind of nuance with the confirmed data to exist at the time.

It's not easy for a obsession, let-alone a career to fall into.

[Anonymous No.10804359]

>>10804344
>I'm around 14 years
That's a bannable offense.

[xxXSuiseisekiXxx No.10804392]

>>10804359
Fuck..
Guess it's time to seek an accomplice.
If Berenstain bears changes back again to Berenstein then all is well.
>fuck me it for reals changed. It was change to stien that was issue to begin. As it was logically correct as authors name.
>but the kids in the read whatever but illustrated books because their comprehension of reading was sound pointing out that it was stain firstly that made teaches question their own sanity.

[Anonymous No.10804439]

What's the difference between a research statement and a statement of intent? The research statement has a word limit of 600 words but the statement of intent does not. This is for a masters grad school application.

[xxXSuiseisekiXxx No.10804455]

>>10804439
>research statesmen
Begging for money from those that want their tax deduction to also be a pissing contest.
So you're to word it as if inbreeding is unknown to the recipient, as they're never seen new money.

>statement of intent.
This is where you can go ham with field specific psuedo words and references to things only you get.
It's still essentially begging for money. But in this case, you're begging for the money that the inbred tax cut 'promised' you. But now you've got to state your case that you should lead. As everyone else is retarded for not coming up with this idea and method.

[Anonymous No.10804459]

>>10803644
Okay!

[Anonymous No.10804460]

>>10804392
Gtfo schizo namefag

[xxXSuiseisekiXxx No.10804604]

>>10804460
What, not even thrown a bone in the form of 'shit I cant even comprehend, so let's see what schzo makes of it'?
Are you that up your own arse as to what you think is academic discordance is. That you think it should be enforced here of all places?
Fucking Christ Anon.
I bet you use new glyph systems instead of giving Western Arabic numerals a new denotation as to not sully them.

[Anonymous No.10804638]

>>10804604
You can't form coherent English sentences, you're using a name on 4channel and you believe in the Mandela Effect. You're fucking retarded.

[xxXSuiseisekiXxx No.10804651]

>>10804638
I suggest you lean about Poe's Law you mongoloid.

[Anonymous No.10804663]

>>10804460
You first need to BTFO someone to then properly use "GTFO".

[Anonymous No.10804756]

>>10804439
A research statement is intended to be more mathematical/precise than a statement of intent, which is usually a fairly fluffy essay.
But this is a question that you need to send to the department you're applying to. They answer application questions all the time, they don't care at all, and that way you aren't relying on some faggot on 4chan to not fuck up your application.

[xxXSuiseisekiXxx No.10804777]

>>10804756
You've got these around the wrong way.
Statements are generalized. Think like Que cards you'd make to talk about an idea to someone not educated past 18, but put linearly to paper.

>fairly fluffy essay

Unless there's a word mandate, don't do this.
You've got to convey confidence in your knowledge and approach.
No piss farting back and forth reiterating the thing you've already said, padding for content.
Treat this as a declaration of war against Autism math X. Make your claim clear, and the standing orders concise.

[xxXSuiseisekiXxx No.10804791]

>>10804777
Oh and mitigations when error is evident is a must in a intent statement*
If you can't show you're aware of pitfalls, and how to fix them. You get shuffled aside as another rote addiction.

[Anonymous No.10804866]

Do different undergrad schools have any more pull/influence on getting into grad school for Pure Math than others?

[Anonymous No.10804985]

I haven't been in this board for long, and I've noticed that there is some rivalry between algebraic geometers and pde's people, and I was wondering why those two specific branches of math.

[Anonymous No.10804991]

>>10804866
Yes.

[Anonymous No.10805013]

>>10804985
there isn't actually any rivalry, i just shout into the void and the algebraic geometers entertain me by calling me a moron

[Anonymous No.10805125]

>>10803971
A sensitive topic?

[Anonymous No.10805154]

Is there a place on the internet to study math?

I just finished my major but I feel empty without solving calculus problems and I feel like a complete moron because I can't solve differential equations

[Anonymous No.10805162]

>>10805154
>Is there a place on the internet to study math?
https://www.coursera.org/browse/math-and-logic
https://www.edx.org/course/?subject=Math
https://ocw.mit.edu/courses/mathematics/

[xxXSuiseisekiXxx No.10805166]

>>10805154
I'm thinking for my own path is to understand function logic at a greater level. Then use 3rd world labor to run the computations.
I assume it'll be wrong at least 1/5th the time. But give a captcha solve is 0.02 cents in the right market. Multi solve requests are probably cost efficient.

[Anonymous No.10805170]

>>10805162

Thanks anon!

[Anonymous No.10805247]

>>10804985
PDE heavy practical and good jobs application on physics.
AG pure math autism few applications.

[Anonymous No.10805296]

>>10804985
PDE's people really want to be mathematicians, but even the existence of algebraic geometrists reminds them that they aren't good enough (intellectually speaking), so they try to lash out.

[Anonymous No.10805364]

>>10804985
It is a meme perpetuated by people that don’t know anything about either field.
Don’t be fooled into thinking that there are actual "algebraic geometers" or "analysts" on this board

[Anonymous No.10805367]

>>10805154
>I feel empty without solving calculus problems and I feel like a complete moron because I can't solve differential equations
Solving "Calculus Problems" and "Differential equations" is about the most boring thing I can imagine.
Maybe try to do some mathematics?

[Anonymous No.10805700]

>Zeidler's Nonlinear Functional Analysis
>5 thick volumes
>average 760 pages per volume
>3800+ pages in total
Good lord have mercy

[Anonymous No.10806066]

>explains everything in a way a brainlet like me can understand

w-wow impressive

[Anonymous No.10806068]

>>10805700
holy fuck

[Anonymous No.10806099]

>>10803942

Graph theory is industry topic more than academic unsurprising

[Anonymous No.10806101]

>>10803971

AG is more of an academic subject so, yes.

[Anonymous No.10806105]

>>10805296

Le ebig baid

[Anonymous No.10806108]

>>10805367
>i need generalizations spoonfed to me because i lack the initiative and creativity to do it myself

Literally every “pure” mathematician.

[Anonymous No.10806115]

>>10806108
Whom are you quoting?

[Anonymous No.10806122]

>BS with distinction in mathematics, Stanford
>PHD Applied Mathematics, MIT
>teach highest level math courses offered at Umass Dartmouth
>collaborate with quantum gravity, astrophy and computational math groups
>Regarded as the chill and wise final boss of UMD by non-brainlets
>Be forced to teach many retard-level college math courses to non-engineers.
>Teaches algebra, discrete, pre-calc at the same high level of understanding and wisdom
>Get disastrous ratings on rate-my-professor from idiots being confronted with the fact they are incredibly terrible at math
Why is the universe so unjust?

[Anonymous No.10806244]

>>10806122
>PhD in applied maths
>he didn't become the department's extremely autistic derived algebraist who can't even remember the product rule
No wonder they make you teach undergrads.

[Anonymous No.10806289]

>>10805364
there are actual analysts on this board, i am one
the algebraic geometers are just sophomores who are excited because they know what a functor is and can recite the definition of a "scheme" as read on nLab

[Anonymous No.10806295]

>>10806122
if you werent such a dick i bet they wouldnt mind as much

[Anonymous No.10806363]

>>10806244
who are you referring to exactly? AH?

[Anonymous No.10806368]

>>10806363
i have a good idea for who but there must be someone like that in most math departments so i'm likely wrong

[Anonymous No.10806374]

>>10806368
it's just that your example of "not remembering the product rule" seems oddly specific. I'm just curious, didn't think there would be someone else familiar with UMD so quickly

[Anonymous No.10806390]

>>10806363
>AH
Jesus Christ that actually made you think of someone. Absolutely laughing to death here.
>>10806374
Nah, relax. I've always had absurd luck for this stuff.

[Anonymous No.10806400]

>>10805162
Those are great recommendations, but I was talking about a site where you can solve problems and discuss the solution with other people.

[Anonymous No.10806403]

Reminder that none of the math people care about is pure math. Study applied math.

[Anonymous No.10806464]

>>10806400
>Those are great recommendations, but I was talking about a site where you can solve problems and discuss the solution with other people.
You can do so on the first two links.

[Anonymous No.10806594]

How can I find a closed form for the coefficients of x in a polynomial such as:

[math]\sum\limits_{0\leq i \leq n}2^{n-i}(x-1)^i=\frac{x(x-1)^n-(x-1)^n-2^{n+1}}{x-3}[/math]?

I derived this sum using basically the principle of inclusion exclusion, so I want to know the coefficients of x^n in terms of n and the index of the sum, and I know that they have to be integers

this sum is hypergeometric, so I think there would be ways to deal with it using a computer if it didn't depend on n.

[Anonymous No.10806596]

>>10806108
???
What are you trying to say?

What exactly is interesting about doing what a computer is a billion times better at than you?

[Anonymous No.10806653]

>>10806594
have you tried expanding (x-1)^i and reversing the order of summation?

[Anonymous No.10806670]

>>10806653

Impossible because the exponent of (x-1) is i, so the inner sum has limits which are a funciton of the index variable of the outer sum

[Anonymous No.10806697]

>>10806670
>>10806594
Here are the coefficients [math]A_m[/math]:
[math] A_0 = \sum\limits_{j=0}^{n} C(j,j;n)[/math]
[math] A_1 = \sum\limits_{j=1}^{n} C(j,j-1;n)[/math]
[math] A_2 = \sum\limits_{j=2}^{n} C(j,j-2;n)[/math]
...
[math]A_m = C(n,0;n)[/math]

Where..
[math]C(i,k;n) = 2^{n-i} (-1)^k B(i,k)[/math]

Where B(i,k) just just the binomial coefficients .

[Anonymous No.10806702]

>>10806594
Why do you think it has a closed-form expression?

You can use the binomial theorem to convert (x-1)^i to a sum of monomials, then change the order of summation to get each coefficient (of x^k) as a sum of 2^(n-i)*C(i,k)*(-1)^(i-k) terms. No idea how (or rather, if) you'd get a closed-form expression for that sum, though.

[Anonymous No.10806705]

>>10806670
>>10806594
Here are the coefficients [math]A_n[/math]:
[math] A_0 = \sum\limits_{j=0}^{n} C(j,j;n)[/math]
[math] A_1 = \sum\limits_{j=1}^{n} C(j,j-1;n)[/math]
[math] A_2 = \sum\limits_{j=2}^{n} C(j,j-2;n)[/math]
...
[math]A_n = C(n,0;n)[/math]

Where..
[math]C(i,k;n) = 2^{n-i} (-1)^k B(i,k)[/math]

Where B(i,k) just just the binomial coefficients .

[Anonymous No.10806707]

>>10806244
based

[Anonymous No.10806711]

Anyone recognize this book?

[Anonymous No.10806712]

[Anonymous No.10806719]

>>10806594
Is it
[eqn]
\begin{cases}
\frac{1}{3} \left((-1)^n+2^{n+1}\right) & k=0 \\
3^{-k-1} \left(3^k (-1)^{k+n} \binom{n}{k-1} \, _2F_1\left(1,1-k;-k+n+2;\frac{1}{3}\right)+3^k (-1)^{k+n} \binom{n}{k} \, _2F_1\left(1,-k;-k+n+1;\frac{1}{3}\right)+2^{n+1}\right) & k\geq 1 \\
\end{cases}
[/eqn]
??

[Anonymous No.10806720]

>>10806594

[Anonymous No.10806727]

>>10806711
McDuff and Salamon, Introduction to Symplectic Topology, maybe.

[Anonymous No.10806754]

>>10806719

I don't even know

[Anonymous No.10806767]

>>10806670
So what...

The binomial theorem is:
[math]<math>(x+y)^n = \sum\limits_{k=0}^n {n \choose k}x^{n-k}y^k[/math]

Just replaced y with -1
and [math]i[/math] with n.

[Anonymous No.10806786]

>>10806727
Doesn't seem to be it. A good book nonetheless though.

[Anonymous No.10806798]

>>10806711
where did you find the image?

[Anonymous No.10806807]

>>10806705

I think you win this

>>10806702

It might not have a closed form expression, but I know from how I derived this that whatever the answer is they must be positive integers.

"Closed form" means "Doesn't have a sigma in it." For my purposes, having a sigma is perfectly fine

>>10806767

I know. You can't put the inner sum outside of the outer sum if the upper index of the inner sum is the variable of the outer sum. Which is what "changing the order of the sum" means

[Anonymous No.10806809]

>>10806798
https://www.ipp.mpg.de/4105128/gspm

[Anonymous No.10806824]

Academic science under capitalism in a nutshell >>10803659

[Anonymous No.10806831]

>>10806807
>You can't put the inner sum outside of the outer sum if the upper index of the inner sum is the variable of the outer sum. Which is what "changing the order of the sum" means
That's find.
The next step would be just to expand the sum a little to see if you find a pattern. The terms of the sum are going to depend on two dummy indices, which is fine. Just set the coefficients (which depend on the two dummy indices and n), to some other symbol like C(I,k;n) to make it easy. Then you expand the sum like this
C(0,0;n)x^0 + C(1,1;n)x^0 + ... +C(n,n;n)x^0
+ C(1,0;n)x^1 + C(2,1;n)x^1 + ... + C(n,n-1;n)x^1
...
+ C(n,0;n)x^n

And work from there.
This sum is just an n-th degree polynomial.

[Anonymous No.10806834]

> applied just for 70 positions
> should have applied for 700

Recruiting discredited to the bone.

It's all fucked up. I remember my story. Applied for 100+ positions. NONE worked. Only got invited to a Skype once.

Ended up landing a position being asked at some meeting by pure accident.

Still have it and have to hold on to it and accept the shittiness of being a professor's slave. No option to jump off.

Recruiting is realy fucked up. Hell, I experienced that first hand. I tell you it's fucked up

[Anonymous No.10806836]

>>10806824
Graph theorists belong in the Gulag.

[Anonymous No.10806841]

>>10806824
Ironically, the capitalist industry wanted him apparently.
So capitalism seems to be working fine, just academia not, which isn't exactly to be known to operate on capitalist principles.

[Anonymous No.10806845]

>>10806836
Don't be triggered by that and don't fall into deception that it's somehow way better in other field.
I am not a graph theory guy myself, but I have enough mentaility to realize it's all scammed disregarding the field.

[Anonymous No.10806852]

>>10806841
> So capitalism seems to be working fine
> Made a conclusion based on a single application by a single dude
> Considers himself worthy of writing in /mg/

In fact, the situation with recruiting in industry is also screwed. Here, in Austria, you need to work 3-4 internship for free/minimal monies at a company to just a chance of getting a position.

[Anonymous No.10806941]

>>10806374
i didn't make the original post. the person i was thinking of has the initials MW. honestly, anyone who knows of him would know the name immediately from that. sigh.

[Anonymous No.10807000]

>>10806836
:(

[Anonymous No.10807165]

>>10803942
>his PhD is graph theory

[Anonymous No.10807189]

How would you prove this statement?
x>y <=>-x<-y

[Anonymous No.10807194]

>>10807189
I wouldn't.

[Anonymous No.10807204]

>>10807189
For what definition of >?

[Anonymous No.10807209]

>>10807204
example
3>2 <=> -3<-2
how can i prove it for every real number?

[Anonymous No.10807211]

>>10807189

depends what you allow x and y to be.

[Anonymous No.10807218]

>>10807209
Show that a>0 implies 0>-a, proceed to exhaust.

[Anonymous No.10807220]

>>10807189
For the satement: for a > b to be true, then for all a,b, a-b is not zero and a - b is positive.
For the statement a < b to be true, then for all a,b a-b is not zero and negative.

If x > y ,then x - y is not zero and positive.
If x-y is not zero and positive, then (-1)x - (-1)y is not zero and negative.
If (-x) - (-y) is not zero and negative, then -x <-y.

Therefore, if x > y then -x < -y.
Using symmetry arguments, swatting x and y, the reverse statement is also true.

[Anonymous No.10807232]

>>10807220
A true >>>/lit/-level proof right here.

[Anonymous No.10807235]

>>10807211
By convention, "<" and ">" are relations defined only for real numbers.

[Anonymous No.10807572]

>>10806807
> I know. You can't put the inner sum outside of the outer sum if the upper index of the inner sum is the variable of the outer sum. Which is what "changing the order of the sum" means
[math]\sum_{i=0}^n \sum_{j=0}^i x_{i,j} = \sum_{j=0}^n \sum_{i=j}^n x_{i,j}[/math]
Either way you get all i,j pairs where 0<=i,j<=n, j<=i.

In this case, changing the order of the sum lets you factor out x^k from the inner sum.
[eqn]
\sum_{i=0}^n 2^{n-i}(x-1)^i
= \sum_{i=0}^n 2^{n-i} \sum_{k=0}^i {i \choose k} x^k (-1)^{i-k}
= \sum_{k=0}^n \sum_{i=k}^n 2^{n-i} {i \choose k} x^k (-1)^{i-k}
= \sum_{k=0}^n \left( \sum_{i=k}^n 2^{n-i} {i \choose k} (-1)^{i-k} \right) x^k
[/eqn]

[Anonymous No.10807589]

>>10807572
this

[Anonymous No.10807669]

>>10803629
ok anons. you know when that time when algebra was the hardest shit in the world? but you come back next year and you are confused why that was ever difficult? what the fuck happened? has anyone written on this?

>>10804308
yes with my bf

>>10803681
depends what you mean by 'moving to physics'

[Anonymous No.10807724]

>>10807669
>you know when that time when algebra was the hardest shit in the world?
No. We were born mathematicians.

[Anonymous No.10807754]

>>10807189
add -x-y to both sides

[Anonymous No.10807755]

>>10807669
algebra is a joke

[Anonymous No.10808083]

>>10806852
I replied to a person who wrote "recruiting in a nutshell" indicating that he believed that this is what commonly happens, going from that I deduced that the capitalist system works much better than academia.

>Here, in Austria, you need to work 3-4 internship for free/minimal monies at a company to just a chance of getting a position.
Press X to doubt.

[Anonymous No.10809192]

>>10807669
>ok anons. you know when that time when algebra was the hardest shit in the world?
Just reduce everything to the study of matrix groups

[Anonymous No.10809341]

>>10804000
hell yeah

[Anonymous No.10809863]

Gromov a best.

[Anonymous No.10809870]

>>10809341
>NIGGA
Why the racism?

[Anonymous No.10810016]

>>10809192
based

[Anonymous No.10810019]

>>10809192
Thats called representation theory.

[Anonymous No.10810167]

Does noetherian induction imply peano induction?

[Anonymous No.10810332]

>>10810016
On what?

[Anonymous No.10810541]

>>10810167
>Does noetherian induction imply peano induction?
What have you tried?

[Anonymous No.10810624]

>>10806727
Duff and Simpson

[Anonymous No.10810627]

>>10810541
What do you mean by tried?

[Anonymous No.10810761]

>>10810627
What do you mean by "mean"?

[Anonymous No.10810796]

>>10810627
>>10810761
what does this symbol mean: "?"

[Anonymous No.10811059]

>>10810796
Why the nazi imagery?

[Anonymous No.10811333]

>>10811059

Sir, can you please refrain from vaping in the mathematics library?

[Anonymous No.10811336]

>>10811333
We're not a "sir".

[Anonymous No.10811621]

For an irreducible affine variety X, can we have a regular function f that is zero in a neighborhood of some point in X but not zero on X?

Surely we can have an irreducible closed set that contains other irreducible closed sets since every irreducible closed set will contain points, which are irreducible closed sets.

Does this fail because it is impossible for an irreducible closed set to contain an irreducible closed set large enough to contain a neighborhood?

[Anonymous No.10811626]

>>10811621
It is not possible for the reason you givr. In an irreducible variety, open sets are dense so, if a function is zero on a neighborhood of a point, then it has to be zero everywhere

[Anonymous No.10811696]

>>10811621
>neighborhood
If it zeroes on a set, this set is closed.
If it zeroes on an open set, we have a set that's both closed and open.
IIRC, the only sets that are both closed and open in an affine variety are the entire variety and the empty set, but I *could* be wrong.

[Anonymous No.10811716]

>>10811696
>IIRC, the only sets that are both closed and open in an affine variety are the entire variety and the empty set, but I *could* be wrong.
Is an affine variety necessarily connected? If so, then you remember correctly.

[Anonymous No.10811729]

>>10811716
To be entirely honest with you, I can never entirely remember the definitions in algeo.
But if a subvariety counts as affine, taking two parallel lines in R^2 gives an easy counterexample.

[Anonymous No.10811799]

>>10811336
>using contractions

[Anonymous No.10812333]

>>10811799
That does not appear to be a well-formed sentence. Please consider adjusting your style to adhere to commonly upheld /mg/ standards.

[Anonymous No.10812346]

>>10811716
irreducible => connected

[Anonymous No.10812360]

>>10811799
>>using contractions
Who are you quoting?

[Anonymous No.10812394]

>>10803681
going to math from physics would probably be easier than the other way around - physicists also tend to excel in all other fields, but the other direction is seen more rarely. Both are rare, really.

[Anonymous No.10812453]

>>10803681
The way that mathematicians and physicists think is entirely different. I aced all of my classes throughout a math major but really struggled in my required physics courses.

Physicists think of things in terms of... well.. physics. They don't think about the rigor as much as they think about what it is that's happening behind the symbols.

Mathematicians build an intuition that is based entirely on rigor and do their damndest to remove any sense of intuition they had before learning how to do real math.

[Anonymous No.10812500]

>reading munkres
>this function is continuous
>ha, the proof must be trivial
>it isn't
wtf James

[Anonymous No.10812514]

>>10804061
>>10804091
>>10804166
>NOOOO REDDIT BAD

[Anonymous No.10812600]

>>10812500
it was trivial. all of undergraduate topology is trivial.

[Anonymous No.10812603]

>>10803629
if one has just finished linear algebra what is next?

[Anonymous No.10812621]

>>10812600
>all of undergraduate topology is trivial.
graduate too

[Anonymous No.10812623]

>>10812603
>if one has just finished linear algebra what is next?
Algebraic geometry.

[Anonymous No.10812628]

>>10812603
Abstract algebra, Topology, and Real analysis are the big three that you'll want. Just pick one. I'd suggest learning the fundamentals of them in that order but picking one of them and running with it won't be a bad idea.

[Anonymous No.10812629]

>>10812603
depends. did you do multivariable calculus?
if not, do that.
if so, then you have a few options.
depending on what sort of linear algebra you did, you could do a more proofy version. that would be axler's linear algebra done right. alternatively you could do some basic algebra, something for groups/rings/fields. if you've never done any proofs before then you would probably find Fraleigh most comfortable, but if you are more motivated/experienced try Dummit and Foote or Artin. to be honest you should really try Artin cause it will also give you some proof based linear algebra.
if you want to learn more about the foundations for calculus and why all that shit works, pick up analysis instead. tao's analysis 1 and 2 are great, also abbott's understanding analysis. pugh's real analysis is a bit tougher but really fun and has amazing exercises. rudin's principles of analysis is pretty dry and tough to swallow, but is widely considered a "rite of passage" book. i'd find a PDF for tao's analysis 1 and a PDF for artin's algebra, and start reading them both. pick the one you like.

[Anonymous No.10812631]

>>10812628
>topology before real analysis
the second chapter of any real analysis book, about metric space topology, should be the way anyone learns topology to begin with. analysis provides the motivation for topology.

[Anonymous No.10812668]

>>10812629
>depends. did you do multivariable calculus?
no
>depending on what sort of linear algebra you did, you could do a more proofy version. that would be axler's linear algebra done right
Axler was the text that I used, I felt like there were some deficiencies because he avoided the use of matrices and determinates for so long and so I filled in the perceived gaps with youtube lectures and skimming other lin alg books recommended in the wiki but I wanted to go over Lax and to work through some of the exercises in the more interesting sections.

I was recommended Zorich, Royden, Rudin, Apostol for Analysis so I suppose I'll focus on that but in general what motivated my interest in lin alg was possibility of applying it to analysis, and other related subjects like probability and measure, and physics.

[Anonymous No.10812676]

>>10812668
oh well if you haven't done any multivariable i'd recommend the first few chapters of spivak's calculus on manifolds maybe? just go for analysis. check out rudin or pugh. pugh is the same level of difficulty as rudin, just more fun and less autistic. you can then do topology, measure theory, and functional analysis afterward (functional analysis is probably what you want if you want to combine lin alg and analysis) or you can do differential geometry, the other big lin alg / analysis thing. but for that you'll definitely want multivariable, and again i'd highly recommend spivak calculus on manifolds for anyone wanting an excellent treatment of some basic diff geo and all the multivar you'd ever want.

[Anonymous No.10812683]

>>10812676
Thank you. I've heard from lurking this general and looking up and very briefly skimming Spivak's Calculus on Manifolds that Munkres might be superior or that Spivak is sort of austere/sparse, and leaves something to be desired, is that founded and is it worth substituting for Munkres?

[Anonymous No.10812690]

>>10812603
>if one has just finished linear algebra what is next?

[Anonymous No.10812692]

>>10812683
sure go for munkres, i'm sure it's equally good. munkres is great usually, i'm just more familiar with spivak.

[Anonymous No.10812698]

>>10812690
the fuck is wrong with the dates in that image

[Anonymous No.10814677]

>>10812603

quadruple integrals and Parker functions

[Anonymous No.10814689]

>>10814677
lol

[Anonymous No.10814874]

Do either of these two facts have a name?

[math]f[/math] is a single variable polynomial in x.[math]f(x)-f(c)[/math] is divisible by [math](x-c)[/math].

The taylor expansion of [math]f[/math] is an identity for [math]f.[/math]

[Anonymous No.10814877]

math isnt science

[Anonymous No.10814885]

This is not exactly trivial because F is a multivariate polynomial instead of a polynomial in just x.

What does [math]F^{(i)}[/math] mean? Is it a partial derivative, or perhaps the sum of the partial derivatives?

[Anonymous No.10814889]

>>10814885

[Anonymous No.10814958]

>>10814889
>>10814889
Well it is the i-th differential of F at x, an i-linear form, evaluated at (T-x, ..., T-x).
In practice, it can be written [math]\displaystyle \sum_{k_1+\dots+k_N = i} \frac{\partial^if}{\partial x_1^{k_1} \dots \partial x_N^{k_N}}(x)(T_1-x_1)^{k_1} \dots (T_N-x_N)^{k_N}[/math]

[Anonymous No.10815039]

tell me, /mg/, what is your optimal study space like? do you prefer using PDF/computer or actual books? desk or lap? cool or warm room?

[Anonymous No.10815062]

>>10815039
I like actual paper. Computers tend to facilitate procrastination in my case so I print everything I need: If I’m reading a book, either I borrow the actual book or I print the sections I want. For articles, I just print whatever I find interesting (it’s free at the office). I like to work at a desk, but if I’m thinking about something I can go for a walk or take a bus ride with a recorder/pen and pad

[Anonymous No.10815200]

>>10814877
But it is a part of "Science and math", the subject matter of the 4chan board /sci/.

[Anonymous No.10815215]

>>10815039
A large Desk with a piece of paper in front of me.
Preferably material related to the subject matter printed out, if that is not feasible a Laptop behind the paper where the relevant documents are displayed in a tilling window manager to minimize distraction.

[Anonymous No.10815532]

>>10807755
category theory is hard! fun but hard

[Anonymous No.10815705]

>>10812603
linear real analysis

[Anonymous No.10816055]

>>10812600
Give any working mathematician a true/false exam on point set topology and I guarantee you they'll fail.

[Anonymous No.10816090]

>>10816055
>[math]T_{-1}[/math]
Do topologists really?

[Anonymous No.10816938]

Why don't more people study boolean functions?

There are important unsolved problems such as enumerating bent functions and the cusick cheon conjecture

[Anonymous No.10817021]

>>10805700
Nice
Just downloaded and added to my collection of books I'll open once every few years and never read more than 5 pages

[Anonymous No.10817939]

>>10815039
i dont mind switching between paper and laptop. i love working outside in the sun or under something while it's drizzling. i also really enjoy working on my lap when i'm just doing casual shit. if i'm trying to be very productive i'll work at my dining table cause usually there's a roommate/person there to keep me productive. i prefer it to be on the colder side and to wear a light sweater, but really i just want the windows open to have whatever the natural air is. i love lots of natural light and such.

[Anonymous No.10817943]

>>10816055
i was joking about topology being trivial, but yeah this shit is nonsense
i have the book "counterexamples in topology" by steen and seebach and i really need to go through more of it because some of those spaces are kino

[Anonymous No.10818307]

im goint to ask this here:

since every formal system is defined by a few defined things: like alphabet, grammar etc, is it possible to enumerate every possible formal system? ie walk through every single one. are they countable somehow?

[Anonymous No.10818412]

>>10818307
Enumerate all strings of symbols. Discard the strings which don't represent a valid formal system.

[Anonymous No.10818673]

>>10818307
Refer to >>>/lit/.

[Anonymous No.10818680]

>>10815039
behind your mother/sister/wife/gf’s pelvis

[Anonymous No.10818695]

>>10806720
Shizo

[Anonymous No.10818976]

I'm halfway through Stroud's Engineering Mathematics and starting Uni to do CS at the grand old age of 26 in September.

Will the book get me through the course, or is there any CS related maths that isn't in the book?

[Anonymous No.10819015]

As of right now, and ever since 1930 (guess why), so called "pure mathematics", a branch of formalism effectively, is a rotting, maggot-festering corpse. Exactly the same way, philosophy became a rotting, maggot-festering corpse in 1740s (guess why).

And yes, I just called all the contemporary pure mathematicians, as well as all the contemporary philosophers, maggots. Which is all they really deserve.

[Anonymous No.10819077]

There is something I don't understand about Gödel's first incompleteness theorem.
Let's take a consistent theory [math]F[/math] in which the incompleteness theorem applies, and its corresponding Gödel's sentence [math]P[/math], which can be understood as "I am not provable inside [math]F[/math]. [math]P[/math] is neither provable nor disprovable in [math]F[/math], hence it is "true but not provable".

Is [math]F + P[/math] consistent?
I would say yes, with the following proof by contradiction :
If [math]F, P \vdash \bot[/math], then [math]F \vdash P \rightarrow \bot[/math], which means that [math]F \vdash \neg P[/math], which is not possible because the incompleteness theorem states that [math]F \not \vdash \neg P[/math]

Is [math]F + \neg P[/math] consistent?
I would say yes, with the following proof by contradiction :
If [math]F, \neg P \vdash \bot[/math], then [math]F \vdash \neg P \rightarrow \bot[/math], which means, in classical logic, that [math]F \vdash P[/math], which is not possible because the incompleteness theorem states that [math]F \not \vdash P[/math]

Therefore both [math]F + P[/math] and [math]F + \neg P[/math] are consistent, but somehow [math]P[/math] is considered "true" (and [math]\neg P[/math] is considered "false").
Am I wrong somewhere?

[Anonymous No.10819108]

>>10818976
> Engineering mathematics
> CS
Not mathematics, please refer to >>>/lit/ or >>>/g/

[Anonymous No.10819115]

>>10819015
Coping CSlet
(Or physishit)

[Anonymous No.10819152]

>>10819077
>Gödel's first incompleteness theorem
Refer to >>>/lit/.

[Anonymous No.10819154]

>>10819015
>formalism
>philosophy
Refer to >>10819152.

[Anonymous No.10819155]

>>10819152
>>10819154
Epic meme, bro!
Did you invent it yourself?

[Anonymous No.10819158]

>>10819155
>bro
We are not a "bro".

[Anonymous No.10819159]

>>10819158
have sex

[Anonymous No.10819161]

>>10819159
Not him or her, but with whom?

[Anonymous No.10819180]

>>10819159
>sex
Refer to >>>/soc/.

[Anonymous No.10819182]

Need help with a problem!
On a table there are 99 red matches, 100 yellow and 101 black ones. In one move you take two matches of different color and replace it with one match in the third color, (ex take yellow, red. Add one black.) After a certain amount of moves there's only matches in one of the color left. Which one?

I've got that the color is yellow but I can't prove why. Please help!

[Anonymous No.10819187]

>>10819182
>On a table there are 99 red matches, 100 yellow and 101 black ones. In one move you take two matches of different color and replace it with one match in the third color, (ex take yellow, red. Add one black.) After a certain amount of moves there's only matches in one of the color left. Which one?
What have you tried?

[Anonymous No.10819191]

>>10819182
my guess is that 101 and 99 are both odd, but 100 is even.

[Anonymous No.10819193]

>>10819182
Consume one yellow and one black to produce one red.
Now you have 100 red, 99 yellow, and 100 black.
Consume 100 red and 100 black to produce 100 yellow.
Now you have 199 yellow.

[Anonymous No.10819202]

>>10819193
Moreover, you can check that, whatever you do, the number of red and black have the same parity, which is different from the parity of the number of yellow. Therefore yellow is the only remaining color possible.

[Anonymous No.10819215]

>>10819202
yep thanks I just wrote some arrangements and thought that must be the case, just had some problem with the argumentation but now I'm good thanks!

[Anonymous No.10819259]

>>10819191
99 isn't fully odd though.

[Anonymous No.10819293]

>>10803629
>2^69 dimensional tic-tac-toe edition
My guess would be it would be an obvious win for the first player due to the sheer numbers of possibilities that need to be defended.

[Anonymous No.10819314]

>>10819293

Ramsey theorists know that in higher dimensions, it can't be a cat game.

There is elementary proof using a strategy stealing argument that the second player cannot have a guarenteed winning strategy.

So yeah pretty much

[Anonymous No.10819369]

>>10819182

A brufe force proof:

If you can do 200 moves, you have removed 200 matches so all piles are empty. Therefore, it is possible to prove that by checking every possible sequence of moves below 200.

A proof that makes use of a homomorphism:

The red and black piles have the same parity initially. Every move changes the parity of every pile. The game cant end while they're both odd since that implies we can still do another move. Therefore, the only possible way the game can end is if yellow is the only one that is odd.

[Anonymous No.10819713]

What is meant by [math]\mathcal{m}_x^2[/math]?

I just don't even....

[Anonymous No.10819752]

>>10819077
>but somehow P is considered "true" (and ¬P is considered "false").
because it literally is true ie the godel number actually has the number theoretic property that he claims it has

[Anonymous No.10819785]

>>10819713
It's just the product ideal m_x m_x, i.e. the ideal generated by products ab where a and b are both in m_x.

[Anonymous No.10819796]

>>10819785

ok thanks senpai

[Anonymous No.10819902]

Math undergrad here
Doing my thesis on hyperbolic geometry on the hyperboloid, but need a specific subtopic to write about.

[Anonymous No.10820246]

>>10818307
Yes they are countable.

[Anonymous No.10820538]

>>10819902
be sure to let us know what you pick!

[Anonymous No.10820705]

Does anyone have good video resources on number theory? I'm reading 'Introduction to the Theory of Numbers', if that matters.

[Anonymous No.10820712]

Any recommendations for studying for quals?

Only two exams, Algebra and Analysis, of course nothing on what I'm actually good at, Topology and Geometry (i.e. I'm not actually good at math I just have +1 std dev spatial intelligence)

Algebra is pretty straightforward but Analysis never is, so that's the one I'm worried about.

[Anonymous No.10820797]

>>10820705
number "theory" is not mathematics
>>10820712
do plenty of quals. time yourself. notice patterns.

[Anonymous No.10820875]

>>10820712
Do a fuckton of problems. Like at least 2-3 a day. And focus on the ones that you don't know how to do or involve some technique you're not comfortable with. Do them until you've internalized every result/technique in your textbook.

[Anonymous No.10821113]

>>10819902
>>10820538
see thats the problem, I have no idea what to pick, I'm interested in determining orbits in space with hyperbolic geometry, but have no idea what to actually write my thesis about.

[Anonymous No.10821706]

>>10820797
>number "theory" is not mathematics
define "mathematics"

[Anonymous No.10821716]

>>10820712
>+1 std dev spatial “intelligence”
>>10820875
don’t help him faggot

[Anonymous No.10821753]

>>10821716
>faggot
Why the homophobia?

[Anonymous No.10821758]

>>10821753
faggot

[Anonymous No.10821762]

>>10821758
>faggot
>>>/pol/

[Anonymous No.10821768]

>>10821762
>defending faggots
did you know they have significantly higher incidence rates of pedophilia, schizophrenia, suicide, drug addiction and anti-social personality disorder?

Really makes you think!

[Anonymous No.10821782]

>>10821768
not mathematics
refer to >>>/v/

[Anonymous No.10821789]

>>10821782
>crusading for a pedophilic, mentally ill, invasive political faction on a Bahraini old world monkey dealing forum is mathematics
>>>/r/eddit
>>>/lgbt/
>>>/out/

[Anonymous No.10821856]

>>10821768
Please don't say "faggot" on /mg/. That's frowned upon.

[Anonymous No.10821864]

>>10821856
please don’t encourage narcissistic undergrads blogging about themselves that’s generally anti-social behavior

[Anonymous No.10821869]

>>10821113
alright, well... do tell us when you decide!

[Anonymous No.10821911]

>>10821758
>>10821768
>>10821789
>>10821864
hey dude i get that you're really triggered right now but this isn't your safe space and honestly it's kind of pathetic that you cry so much about what people do in their private lives
in any case you're derailing this thread because someone responded to you with a common /sci/ meme and you continue to take the bait
it's clear you've been on this board for at most a week or two, and i don't think it's uncouth for me to tell you to lurk more.
see you soon!

[Anonymous No.10821978]

In algebraic geometry, what is meant by "a linear form on a set O"?

Does that mean a linear equation which is satisfied on that set? Or is it a function which has the algebraic property of being linear (i.e. being a vector space homomorphism)?

[Anonymous No.10822006]

Has anyone read the last Tooker
I think he's slowly starting to break down even when he tries to be serious

[Anonymous No.10822021]

>>10822006
Someone should redirect him to /lit/.

[Anonymous No.10822257]

Was it kino?

[Anonymous No.10822262]

>>10822257
Unless someone animated it, no. It's a book.

[Anonymous No.10822307]

>>10822262
Ok autist

[Anonymous No.10822557]

>>10822257
Category theory is autism.
>>10822307
Ironic.

[Anonymous No.10822616]

>>10822557
>ironic

[Anonymous No.10822633]

>>10822257
>Was it kino?
Categories for the Working Mathematician is a classic non-rigorous book.

[Anonymous No.10822723]

>>10822633
Wait, hold up.
If CFTWM is non-rigorous, what the fuck is p-adic Teichmuller theory?

[Anonymous No.10822733]

>>10822723
can you just go back to fucking art of problem solving already?

[Anonymous No.10822752]

>>10822723
>If CFTWM is non-rigorous, what the fuck is p-adic Teichmuller theory?
Rigorous.

[Anonymous No.10822760]

>>10822021
This is a bad meme since /lit/ is actually a good board

[Anonymous No.10822809]

>>10822723
Could you refrain from the needless profanity?

[Anonymous No.10822840]

>>10822809
Where the fuck do you think you are

[Anonymous No.10822851]

>>10822840
On a respected thread unsuitable for such poor form.

[Anonymous No.10822881]

>>10806852
>In fact, the situation with recruiting in industry is also screwed. Here, in Austria, you need to work 3-4 internship for free/minimal monies at a company to just a chance of getting a position.
Do you have diarrhea? Because that’s a lot of bullshit coming out of your ass.

[Anonymous No.10822901]

>>10822760
>good
>bad
These are meaningless notions.

[Anonymous No.10822942]

>>10803629
lol mods deleted my posts for offending the tranny

[Anonymous No.10822967]

>>10822942
>tranny
Why the transphobia?

[Anonymous No.10822987]

>>10803629
Can somebody explain the process of finding roots of higher degree polynomials? For example, I know there are formulas in radicals for roots of polynomials of degree up to 4, and by the Abel-Ruffini theorem, there is no formula for the general polynomial of higher degree... But I also know that the general quintic polynomial can be solved by the inverse of (x^5-x) and radicals. So how did they come up with this. It's been a while since I've done any abstract algebra, so any help is appreciated.

[Anonymous No.10823103]

>>10822840
Please, watch your language.

[Anonymous No.10823109]

>>10822987
They probably thought about it for a good while, and then wrote down a proof. Have you given that a shot?

[Anonymous No.10823170]

whats the difference between algorithm and computation? It seems to me everything algorithmic can be computed and vice versa

[Anonymous No.10823180]

>>10823170
An algorithm is the description of a computation.
A computation is the execution of an algorithm.

[Anonymous No.10823183]

>>10823170
The algorithm is the process of solving a problem (usually, in the context of finding a particular function, or number, see Euler's algorithm for GCD, Eratosthenes sieve for prime number, Rames algorithm for approximation polynomials...
A computation is just applying theorems, algorithms relations and definition to get a result.
In other words, the computation is the execution of the algorithm

>>10823103
>>10822967
Contribute to the thread meaningfully instead of fishing for (you)s retard

[Anonymous No.10823196]

>>10823170
>>10823180
>>10823183
Refer to >>>/g/.

[Anonymous No.10823203]

>>10823183
Please do not use the r-word. There are algebraic geometers in this thread who might be offended that you are using derogatory language which marginalizes them.

[Anonymous No.10823219]

Jesus Christ this board is even more cancerous than I remember

Anyways ; anyone read https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics
What do you think of it

I think Arnold is basically just showing off how much he knows about his geometry in it and trying to flex on French mathematicians without actually giving a real purpose to mathematics.
I wonder how much of it is satire and how much of it he actually believed

4chan thinks my post is spam because of the URL

[Anonymous No.10823334]

>>10823183
Why the ableism?

[Anonymous No.10823375]

>>10823203
>r word
Results?
>>10823219
>mathematics is a part of physics
Stopped reading there.

[Anonymous No.10823425]

>>10823219
>4chan thinks my post is spam because of the URL
One of the rare cases when 4chan is actually right.

[Anonymous No.10823611]

>>10823180
>>10823183
then how on earth do we know non computable numbers exist

[Anonymous No.10823732]

Is it worth it to learn high school competition math? I've always been pretty good at school curriculum math but never did math competitions in elementary, middle, and high school because I wasn't interested. I took AMC12 in junior year and got a mediocre 60th percentile. Am I missing out on anything? Should I go back and try to learn competition math? Does being good at HS competition math benefit your math abilities later on? What if I want to do the putnam exam eventually in university?

I just picked up a copy of AoPS Vol 1 and i'm getting really fucking annoyed that I have to do dull, long calculations simplifying roots and shit. Is it worth it? Also please give tips for learning competition math if your answer is yes, like should I try to do all the problems in AoPS even if they're boring and long and frustrating?

[Anonymous No.10823788]

>>10823732
There is no actual value in learning how to do contest problems when you are too old to write contests, unless you just enjoy doing the problems.
>I just picked up a copy of AoPS Vol 1 and i'm getting really fucking annoyed that I have to do dull, long calculations simplifying roots and shit.
I'm pretty sure you haven't made it past page 10 yet judging by the content you're describing. Chill out a little.
You're in the introduction to the book where they're making sure you know arithmetic. This is not supposed to be (or even possible to be) fascinating material, it's foundational necessities. If they're hard for you to do correctly, then you should be concerned that your foundations suck and you need to practice. If they're too easy, then skip until you hit sections you don't know already.

[Anonymous No.10823996]

>>10823611
well, every computable number is represented by a finite sequence of characters (the program which, when run, produces it)
since there are countably many such sequences, there are countably many computable numbers
thus there are uncountably many uncomputable ones
>>10823732
competition math is essentially worthless, it's exactly what you think. long, boring, and frustrating. pick up a book on real math and learn some real math. that shit isn't relevant for the putnam at all, for the putnam you'll want some basic discrete math, combinatorics, linear algebra, group theory, elementary number theory, and analysis. and you'll want to be very strong with writing proofs. keep in mind that the majority of people get 1 or 0 points out of 120 on the putnam.

[Anonymous No.10824014]

>When you find out skolem's paradox is infinitely more interesting than the incompleteness theorems

[Anonymous No.10824098]

>>10823732
>Is it worth it to learn high school competition math?
Only if you are enjoying it for your own sake.
But being able to quickly solve basic calculus problems really is of very little worth, if you are actually interested in eventually doing research mathematics.

[Anonymous No.10824194]

>>10824014
Gödel's incompleteness theorem implies the existence of non-standard models of arithmetic.
For any undecidable sentence [math]A[/math] in a consistent theory [math]T[/math], both [math]T + A[/math] and [math]T + \neg A[/math] are consistent, which means there is a model of [math]T[/math] where [math]A[/math] is true, and there is another model of [math]T[/math] where [math]A[/math] is false. For instance, if you note PA for Peano arithmetic (we assume PA is consistent), PA + [math]\neg[/math]Con(PA) is also consistent (even though it is a theory which states that PA is inconsistent), and there is a model of PA where Con(PA) is false.
The trick is that Con(PA) means [math]\forall x, \neg {\rm Proof}_{\rm PA}(x,\bot)[/math], and you know that, for any NATURAL NUMBER [math]x[/math], either [math]x[/math] corresponds to a proof (which cannot be a proof of [math]\bot[/math] because PA is consistent), or [math]x[/math] corresponds to garbage, and in both cases, [math]{\rm Proof}_{\rm PA}(x,\bot)[/math] is false. But there are non-standard models with non-standard numbers which do not correspond to anything, so there may be a non-standard integer [math]x[/math] in such a model such that [math]{\rm Proof}_{\rm PA}(x,\bot)[/math] is true and therefore Con(PA) is false.

[Anonymous No.10824358]

I started doing the daily putnam problems recently. The remi poster motivated me to do it.

When does the webpage get updated? I'm still seeing the same problem as yesterday.

[Anonymous No.10825261]

Here is an idea I had today. Consider the following algorithm.

In spherical coordinates, choose an arbitrary point [math]\vec{a}[/math] with r=1, polar angle less than pi/2.

Then, you construct a transformation of R^3 that maps the origin to [math]\vec{a}[/math] and has the property that the original origin and [math]\vec{a}[math] are colinear with the point (r=1, polar angle=0, theta = 0).

Now, you do the same thing in the new coordinates and keep track of the position in the original coordinates.

Does anyone recognize this? I'm sure it has been thought of before. It probably arises as a way to construct a continuous random walk or approximate a function.

[Anonymous No.10825541]

>>10824014
describe it to a layman

[Anonymous No.10825551]

>>10823219
I remember reading some of it.
I don't really agree, some people don't think in that intuitive way and find it more comfortable to reason in abstract terms.

A lot of intuitive thinkers seem to be utterly determined their way of thinking is superior and natural.
I suppose the pendulum swings both ways.

Another thing: every single 'How to Write Mathematics' pdf etc always talks about using words rather than symbols.
But some people like the abstract symbols, which communicate a lot of information and are much easier to read.
I find stuff like Munkres awful to read because it's so wordy.

[Anonymous No.10825558]

>>10825551
if you don't think about math in an intuitive way then you don't like math, you like autism, and you're under the impression that these are the same thing.
you are mistaken.

[Anonymous No.10825593]

>>10825558
You don't own or oversee mathematics, so you shouldn't be telling people what is 'real' mathematical thought and what isn't.

Actually, I agree that the end goal is to be comfortable enough with concepts that you can manipulate and see connections in a way that feels 'intuitive'.
I don't agree that Arnold's ideas are the correct way for everyone to reach that level of understanding.

Mathematics provides a way of describing physical phenomena.
Some people are interested in what is being described, and others are more interested in the description itself. Between this exists a spectrum.

[Anonymous No.10825616]

looking for suggestions on PDEs textbook, requirements:
1.) not evans
2.) not an engineering book
3.) doesn't refer to PDEs as PDE

opinions on:
jürgen jost - partial differential equations
fritz john - intro to PDEs
this -https://doi.org/10.1017/CBO9780511623745
welcome

[Anonymous No.10825618]

>>10825541
There is a countable transitive model of ZFC (assuming it is first-order logic).

What does this mean?
You can think of any axiomatic theory as an algebra, i.e. you have a bunch of axioms, and you prove these axioms are true for some set.

Peano arithmetic, for instance, is an axiomatic theory of arithmetic, and the usual [math]\mathbb N[/math] is a model for it, therefore everything you can prove in Peano arithmetic is true in [math]\mathbb N[/math]. But there are of course plenty of models of arithmetic.

Same thing goes for set theory. There are models of ZFC, i.e. sets which satisfy all the axioms of ZFC. The models of ZFC are of course defined inside ZFC, so you can see it as a russian doll containing itself. Moreover, there are transitive models of ZFC, i.e. models where the "[math]\in[/math]" relation is modelled by itself. In a transitive model, you have the set of integers, intersection of sets is the real intersections, and everything behaves like the real stuff up to a point.

Take a transitive model of ZFC. It is infinite because you have all the integers inside, and it can be uncountable. But the set of theorems of ZFC is countable (because a proof is a bunch of text, which is countable), so the theorem of Löwenheim-Skolem states there are elements of the model you can remove without breaking any of the theorems. In fact you can end up with a countable model of ZFC.

This may seem to contradict the Cantor diagonal argument, because, if everything is countable, then [math]\mathbb R[/math] is countable. Actually, [math]\mathbb R[/math] is said to be uncountable because there is no one-to-one correspondence between [math]\mathbb N[/math] and [math]\mathbb R[/math], which is the case in the countable model: some of the bijections between [math]\mathbb N[/math] and itself have been removed from the model.
Philosophically speaking, it could mean there is only one infinity, and the different infinite cardinals only show the absence of a bijection.

[Anonymous No.10825622]

>>10825616
>PDEs textbook
>not an engineering book
No such thing.

[Anonymous No.10825638]

>>10825622
bore off

[Anonymous No.10825658]

>>10806845
>Graph theory
what do you mean by "its all scammed disregarding the field"?

[Anonymous No.10825664]

>>10825616
partial differential equations by emmanuele dibenedetto

[Anonymous No.10825742]

>it's an intuition is the same as geometry episode
Fuck off, Atiyah. Don't make me call an exorcist.

[Anonymous No.10825766]

>>10823219
French person who went to an ENS in the past. I think there is some exaggeration and some degree of flexing in this text, but I feel like he has pointed out something very interesting and something I have been thinking about a lot during my years there.
Basically, it is true that our classes are very abstract, with few examples sprinkled in. It obviously depends on the professors, but it is the general tendency. We did have tutorials for all our classes which covered examples, including those he mentioned, but I have always felt like it was not enough to gain a deep insight into the math we were doing.
Contrary to my prepa years where we had few tools but did amazing things with them, at the ENS we were learning an amazing array of new tools but over such a short period of time and so few drills/computations that we ended up forgetting most of them a semester/year later.
For whatever reason, it has always seemed to me like russian mathematicians (at least those I have met) had a deeper understanding of math, or at least a much more holistic one, probably due to their much more hands-on approach and their emphasis on the physics behind math.
Even as an algebraist, I regret to this day not having put in more time to study physics.

[Anonymous No.10825787]

http://matematicas.unex.es/~navarro/res/zalamea.pdf

Grothendieck resume math works (spanish)

[Anonymous No.10825822]

>>10825766
>that one time Perelman proved the Soul conjecture in three pages with basic results

[Anonymous No.10825825]

>listen to Atiyah talk
>can't understand a word he says
Anyone else?

[Anonymous No.10825853]

>>10815062
>Computers tend to facilitate procrastination in my case so I print everything I need
Me too. I also tend to have certain spots in my place where I do goof off shit on my laptop and then others where I do actual useful shit.

[Anonymous No.10825856]

>>10818307
Isn't that the premise behind Godel's incompleteness theorems?

[Anonymous No.10826121]

>>10825616
>not evans
why not?
>doesnt say PDE
why the fuck not?
do you have autism? maybe algebra is better for you.

[Anonymous No.10826447]

Tips for getting better at analysis for a graduate student? Or perhaps another way of phrasing it, how do I stop relying on intuition? I can do most of the exercises in topology books like May, Hatcher, Rolfsen, geometry books like Spivak, Petersen with ease, because intuition gets you most of the way there, but in a subject like analysis, which is far less intuitive, where I can't 'see' things, I have a lot of trouble.

[Anonymous No.10826704]

>>10803659
except the guy from this pic already had a PhD

[Anonymous No.10826889]

>>10825664
thanks i'll look into it

>>10826121
>not evans
i tried it and didn't like it, i never seem to get along with any of the AMS books actually

>doesn't say PDE
i'm being partially tongue in cheek but i do find it really really jarring

>do you have autism?
not strong enough to be diagnosed and i developed normally in childhood

[Anonymous No.10826899]

>>10825616
The engineering ones are mostly written by mathematicians anyway. Problems in engineering drive the frontier of the field. No one is working on baby PDE analysis "just because", the only research that pays is dealing with real problems occuring in industry.

The only possible exception to this is in non-linear dynamics in chaos theory which is a suprisingly small field and most work is on ODEs at the moment.

But in any case get that stick out of your ass. It's an applied field, you'll be dealing with engineering.

[Anonymous No.10826903]

>>10803659
Please don't do this. My mental health needs a break.

[Anonymous No.10826908]

>>10803960
What do you mean? Graph theory has a lot of funded positions.

[Anonymous No.10826913]

>>10804079
How do you even apply to more than 50 without just sheer CV spamming?

[Anonymous No.10826934]

>>10819182
Think through this with 1 red, 2 yellow, and 3 black matches keeping in mind the parity of each match pile.

[Anonymous No.10826938]

>>10806101
You kids have no idea how academia works. Academic subjects put you in a worse position not a better one.

Let me paint you a quick picture of research groups in a typical math department:

Cutting edge, pure research groups:
>1-2 Professors with 40 year's worth of publication in the field, 1 postdoc and 10+ grad students

Explanation: The livelihood of these professors mostly depends on the teaching position they have to teach that one upper level undergraduate model and whatever obscure postgrad seminars/modules the department offers to the 1 or 2 students that show an interest that year. Of course, they also have that one grant that NSF throws pennies at once every 10 years to keep academics happy.

Now the question is do you really think that you, a snot nosed PhD graduate with 7 journal papers and 14 citations is going to steal away this professors once in a decade grant competing against his 5000+ citation, Fields medal containing CV while also competing against 1000s of other recent PhD grads in the field?

Applied research groups
>3-5 Professors, 10+ Associate professors, 20 postdocs, 10+ lecturers, 5 grad students

Explanation: Industry grants, countless applied math undergraduate courses to scientists and engineers, justifiable funding from NSF-like sources to use tax money on real world problems. Grant links to engineering, physics and finance departments. Few grad students since graduate pure math students are too snobbish to care about the field.


I'm barely exaggerating in this post. I am exaggerating, but only just.

[Anonymous No.10827444]

>One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry.[1] For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory.

what can I say but "based and redpilled"?

[Anonymous No.10827488]

>>10827444
The entire field of number theory is just one big disgusting application.

[Anonymous No.10827616]

going to reask this question: Why do people on here dislike graph theory?

[Anonymous No.10827626]

>>10823732
This is like saying you want to be a sculptor and deciding to practice using a coloring book.

[Anonymous No.10827628]

>>10827616
Why do you care what anyone here thinks ? Why do you assume they know what they are talking about ? Hell, why do you think they think at all ?

[Anonymous No.10827633]

>>10827628
> Why do you care what anyone here thinks ?
I saw in this thread shit talking graph theory. Never having taken a course in that, but having been curious about it, I wanted some insight from someone who (I presume) has more knowledge in it than I

> Why do you assume they know what they are talking about ?
Because they replied implying something about graph theory when I have barely any opinion on the subject. Perhaps their opinion is malformed, nonetheless I'm curious to know what it is.

> Hell, why do you think they think at all ?
Because I am a thinking person, therefore they must be thinking as well.

[Anonymous No.10827693]

>>10825766
I don't know if this is the Algebra you studied, but space groups (and most importantly, their representation) play some role in Crystallography and Solid-State Physics. However, we are mostly taught the end results without time for the formal proofs or the formal approach ; we're just told that in 1920 they used it to prove that there's only 14 kinds of crystals possible, which is a shame, really.

Group theory and linear algebra are widely used in my own courses (applied physics), but I had to look at the proofs myself.

It's a bit of a shame that the highest I got to study pure math was extremely basic complex analysis (Cauchy integral form, Fundamental Theorem of Algebra proof, basic power series, basic Fourier Analysis...)

Now if I want to learn more advanced maths I need to read books on my own and I'm never quite sure if I understand it right.

[Anonymous No.10827697]

>>10825616
https://www.springer.com/gp/book/9783540404484
It's essentially a mathematical physics textbook though

[Anonymous No.10827698]

>>10827633
I took a basic class that had some graph theory on it. Like really really basic stuff. But it was fun. If I had free time just for fun, I’d sit in on some more classes about. Only thing is that I didn’t get that feeling of feeling smarter I got out of other classes. It was more like a game to me desu.

[Anonymous No.10827736]

>>10827616
There's an idea (which has a kernel of truth inside it) that many mathematicians either consciously or unconsciously view combinatorics as "not real math". This is not remotely as true as it might once have been, but there is still something of a separation between combinatorialists and "the rest" of mathematics. Large discrete math groups are not common, and I've even been to (good) schools with 1 lone graph theorist stranded in a department with not a soul who gives a shit about his work.
The shitposting you read here is just the "not real math" snobbery cranked up to 11 for memes where only scheme-theoretic wank is an acceptable career choice.

[Anonymous No.10827751]

Sorry for the boring question, but what the hell does this notation mean?(Note in case it's not obvious, let V be a vector space, it's V^directsum 2)

[Anonymous No.10827764]

>>10827751
Take the direct sum of 2 copies of V.

[Anonymous No.10827769]

>>10827633
The meme is that discrete math is for computer science, and real math is for physics, and physics RULEZ, and CS fags are dumbz

[Anonymous No.10827779]

>>10827764
Thanks.

[Anonymous No.10827780]

>>10827736

true and based

The average unwashed math major may have never been exposed to combinatorics. They take many years of calculus courses designed for engineers and they don't even have to know what a set or induction is until they take an analysis class in their 3rd year. In their 4th year, a lot of them take another calculus class and then graduate.

It is quite sad, really. They didn't study ramsey theory in highschool and it shows

[Anonymous No.10827791]

>>10827780
I saw induction in high school and this is a very common proof technique at all levels in all fields (Algebra, Analysis, Number Theory...) so I doubt it.
What even is advanced combinatronics, anyways ? I think the real issue is that you basically learn Newton's binomial formula, some boring identities with binomial coefficients, maybe you learn the relationship between arrangements and the symmetric group, about that's all my knoweldge of combinatorics

[Anonymous No.10827802]

>>10827791
>what is advanced combinatorics
The same as normal combinatorics. Arrangements invariant under certain symmetries, geometric arrangements, Haruhi problems, estimates for solutions, bounds, etc.

[Anonymous No.10827804]

>>10803629
Are any anons here good with simplicial sets?

I've heard the claim that the Hurewicz map comes from the prolongation [math]F:\mathbb{sSet} \rightarrow \mathbb{sAb} [/math] of the free abelian group functor. I'm having trouble understanding what is meant by this. The Hurewicz map should be a natural transformation [math]\pi_n(-) \Rightarrow H_n(-)[/math]. I know that for a topological space [math]X[/math], the singular simplicial set [math]Sing(X)[/math] has the same homotopy groups calculated in [math]sSet[/math], and that the homotopy groups of [math]F(Sing(X))[/math] calculated in [math]sAb[/math] are just the homology groups of [math]X[/math]. My guess would be that since [math]F[/math] is part of an adjunction [math](F,U)[/math] for [math]U[/math] the forgetful functor, the Hurewicz map should come from the unit [math]\text{Id} \Rightarrow UF[/math] of this adjunction. I can't see how to make this work though, since applying [math]U[/math] to [math]F(Sing(X))[/math] takes you out of [math]sAb[/math]. Thoughts?

[Anonymous No.10827826]

>>10827791
What >>10827802 said, also Stirling numbers and stuff.
Read "Concrete mathematics" by Graham-Knuth-Patashnik.

[Anonymous No.10827841]

>>10827802
>>10827826
Sounds pretty cool
I wish I had more time and IQ to learn about mathematics as a poor engineering student. I'm trying to read Rudin (having done some analysis "honors" classes in undergrad), and it's pretty hard.
I will add this to my never ending backlog.

[Anonymous No.10827870]

>>10827791

> I saw induction in high school and this is a very common proof technique at all levels in all fields (Algebra, Analysis, Number Theory...) so I doubt it.

In 10th grade, you probably learned about induction in a geometry class modeled after Euclid's Elements. This concept of induction is extremely specialized. The idea of "general induction" is that you prove a statement about an arbitrary construction by showing every given case depends on some known set of cases.

The first two years of calc classes designed for engineers I took didn't require any understanding of it. The only proofs I encountered were like "A vector space is an object defined by a list of arbitrary axioms. Prove some trivial identity using them." or "Set up and solve this integral."

> What even is advanced combinatronics, anyways ?

I think a lot of it could be described as "extremal combinatorics." Things such as ramsey theory, combinatorial design theory, coding theory and other specialized topics

[Anonymous No.10827918]

>>10827870
Induction the way I learned is rather straightforwards
Let P(n) be a property depending on n ; prove the property at rank 0 ; prove that P(n - 1) => P(n) ; then P(n) holds for every n.
I'm French for all that matters.

I only got two semester of calculus in my engineering studies, they were of course mostly concerned with computing integrals but we did get some insight on Riemann sums and Cauchy sequences/epsilon-delta formalism, however we don't really see WHY it's useful if we don't take electives on advanced math.
I did some linear algebra that introduced us to the notion of isomorphisms, relations of equivalence, and congruence classes. I then took an optional module titled "Advanced Analysis" that was more proof-based ; we saw the elementary theory of power series and Fourier series, the construction of real numbers (with Cauchy sequences) and some extremely basic complex integration techniques (residue theorem, Cauchy form, Liouville's theroem, Fundamental theorem of algebra proof...)

This course is basically what motivated me to study math on my own. I'm trying to read Rudin now with the baggage I got from this.

[Anonymous No.10827948]

>>10827918
Induction can be generalized for any well-founded set, see Noetherian induction or transfinite induction.

[Anonymous No.10827953]

Post your favorite inductions.
>show that it works for any prime
>show if it works for n and m it works for nm

[Anonymous No.10827971]

>>10825593
>some people are interested in what’s being described
>others are interested in the description itself
isn’t the second just autism?

[Anonymous No.10827996]

>induction
Isn't that one of those more advanced combinatorial maths? I'm in my 4th year and we haven't went over that yet.

[Anonymous No.10828001]

how do I prove this without l'Hospital rule ? please help frens

[Anonymous No.10828003]

>>10827971
You can illustrate this with the common saying, "it's the journey that matters, not the destination."
In mathematical terms, it's the proof that matters, not necessarily the result itself.

Usually, a proof can help you to understand the result in a way that feels intuitive and can underline the connection of the idea you're studying to another (for instance, you can find a neat connection between matrices and Lesbegue measure if you try show its invariance by translation)
Landau once said that a method was more important than a discovery, because the method can be applied to other problems.

[Anonymous No.10828017]

>>10828001
Hint: limits with an x alone in the denominator are often derivatives of something.

[Anonymous No.10828029]

>>10828001
[eqn]
1 = a^0
[/eqn]
[eqn]
x = (x - 0)
[/eqn]
Hence
[eqn]
\lim_{x -> 0} \frac{a^x - a^0}{x - 0} = \frac{\mathrm{d}a^x}{\mathrm{d}x}(0) = \frac{\mathrm{d}e^{x \ln a}}{\mathrm{d}x}(0) = \ln a * e^{0 * \ln a} = \ln a
[/eqn]

When you have a limit towards 0, with [math]x[/math] on the denominator, always think of the derivative.

[Anonymous No.10828032]

>>10828001
For what definition of ln?

[Anonymous No.10828111]

>>10828029
thank you anon

[Anonymous No.10828211]

>>10807669
be my 2hu tranny bf too pls

[Anonymous No.10828732]

>when you work with a Hausdorff space, what do you visualize?
>when you work with a compact Hausdorff space, what do you visualize?
Personally, R^2 and S^2.

[Anonymous No.10828776]

What do I have if I have a group but without the existence of an inverse. My identity element is still unique.

[Anonymous No.10828817]

>>10828776
>What do I have if I have a group but without the existence of an inverse. My identity element is still unique.

[Anonymous No.10828822]

>>10828817
Thanks my dude.

[Anonymous No.10828824]

>>10828822
>Thanks my dude.
I'm not a "dude".

[Anonymous No.10828840]

>>10828776
>my identity element is still unique

[Anonymous No.10828852]

>>10828840
>assuming left identity is the same as right identity

[Anonymous No.10828891]

How should I think of a sequence space? Is it a set A={a1,a2,...an} of functions , where each aj can be expressed as a function fj(x)? Or is it a set of sets where each aj is a set of scalars?
What then, does the basis look like?

[Anonymous No.10828959]

>>10828891
>What then, does the basis look like?
What have you done to show that it exists?

[Anonymous No.10828998]

>>10828959
I've been asked to prove or disprove that the set composed of vectors ei {e1,e2...} where ei is a vector with with only 0s except at the ith position is a basis. I don't know how it could be but that may be due to my poor understanding of sequence spaces.

[Anonymous No.10829558]

>>10828824
Thanks lad

[Anonymous No.10829714]

>>10819077

You began with "let's take a consistent theory F". That means that in the metatheory you work in you assume Con(F). But Con(F) implies P (this implication can even be proven in a weak system such as PRA). That's why P is considered "true", because it follows from that assumption.

[Anonymous No.10829720]

>>10828891
It's a vector space, where the vectors are sequences. I.e. a set of sequences such that if P and Q are members of the set then so is aP+bQ for any scalars a,b. For any sequences X and Y, (kX)[n]=k(X[n]) and (X+Y)[n]=X[n]+Y[n], i.e. sequences can be scaled and added element-wise. Informally, it's a vector space with an infinite number of dimensions.
>>10828998
Any sequence X=[x0,x1,...] can be written as x0*e0+x1*e1+..., and that representation is unique (in terms of writing X as a linear combination of the ei), so the ei form a basis.