/sci/ - Science & Math

First for fuck Nikolaj.

Elliptic curves? More like elliptic SHIT

>>12130811
I would.

>>12130811
Why

>>12130804
Here's a nice exercise, that I completed yesterday:
Prove that the image of a [math]C^1[/math] curve [math]\gamma:I\rightarrow \mathbb{R}^n[/math] has measure zero wrt to the n-dimensional Lebesgue measure.

If I have a continuous map [math]f: X \to Y [/math] and subspaces [math]A \subset X, B\subset Y [/math] such that f restricts to a homeomorphism [math]X-A \to Y-B [/math]. And A is homotopy equivalent to X, is then B homotopy equivalent to Y?

>>12130860
n >= 2 of course, no?
Maybe there's an issue with the following:
Because the curve is C^1 on a compact interval, it has bounded derivative and thus bounded length (integrate norm of derivative wrt lebesgue on I). Then attach a ball of radius epsilon to every point of the image. The measure of the union of these balls should be no more than the measure of one ball times the length of the curve (this is where i'm uncertain, but i'm sure i can prove this more in detail). Well then the measure of the curve is bounded by C*epsilon^2. Let epsilon go to zero.
I don't know how Sard's theorem does it.

>>12130912
*I mean that A is a deformation retract of X

>>12130938
yes, of course, sorry i forgot

>>12130939
I thought i found a counterexample, but it doesn't work, do we know if it's true or not?

>>12130331
How the fuck did you get this? How the fuck?
I have no memory of that shit.
>>12130912
>>12130954
I also thought I had a few counterexamples and all of them ended up satisfying the result when I thought harder. But I have no intuition for why it should be true. Weird.

>>12130938
Yes, I used Saard's lemma, and the implicit function theorem. If a point is regular, then we can represent f locally as the graph of a function in an appropriate coordinate system.
We know, that the graph of C1 function is measure 0 (since the interval is compact it's Lipschitz, and from that it follows pretty easily).
And now we use a special case of the Saard lemma, and from that we get that the set of critical values are also measure 0.

>>12130938
Yeah, that estimate needs some justification, but it looks true to me as well.

>>12130954
I have no idea if it's true or not. That's why I'm asking

>>12130959
Certainly, I only learned a very smooth version of Sard so I suppose I should look at how much regularity one really needs. I remember the proof being very strongly dependent on things being C^infinity.
>>12130972
My only idea for how to show that estimate is to embed in one dimesnion higher, moving at constant speed foward in the extra dimension, and then there the (n+1)-measure is volume of sphere * length of curve by Fubini. Then you need to project back. I have no recollection of whether such orthogonal projections are monotone for the lebesgue measure though.

>>12130998
Oh, there's no such monotonicity of course. Just take a big set in R^n and give it a very small amount of thickness in the (n+1) direction. Hmm.

>>12130998
Yeah, we learned only the smooth version as well, but if you look at the proof at the last step we can bound the index where the nested sets zero out

>>12131016
I see. Interesting.

>>12131016
Not that this is our case but here's a surprising MO post which I wouldn't have expected.
https://mathoverflow.net/questions/258141/a-counterexample-for-sards-theorem-in-c1-regularity

>>12130860
The curve is C^1, so it's rectifiable.
Assume it has finite length for convenience. Parametrize if by arc length [math]\gamma : [0, l] \rightarrow \mathbb{R}^n[/math]. Cover the curve by placing balls of radius [math]2l/m[/math] at the points [math]\gamma (k l/m)[/math] for integers [math]k[/math] between zero and [math]m[/math], where [math]m[/math] is also an integer.
The volume of the balls decreases like [math]1/m^n[/math] and the number of balls grows like [math]m[/math], so it goes to zero.

Filling in the details and correcting the mistakes is left as an exercise to the reader.

Forgive the Latex, I'm on my phone.

>>12130841
Stupid tranny.

Category theorists get the beating.

>>12131355
*poisoning

>>12130912
No

I take a sphere and identify two points. How should I prove its fundamental group is Z?

>>12131372
have you tried van kampen?

>>12131377
>have you tried van kampen?
No. Should I?

>>12131388
maybe you should try using standard techniques before asking someone else

>>12131390
Maybe. Maybe not.

>>12131401
Maybe fuck yourself.

I started a master program but I don't feel inspired to do anything in particular.
What's the easiest thing I can work on for my thesis?
>just drop it bro
n-no!

>>12131404

>>12131408
openproblemgarden.org
Find a problem to work on here.

bros... i didn't do maths today...

>>12130957
xd

>>12130957
>How the fuck did you get this? How the fuck?
It could just be an approximation obtained by testing different familiar functions and constants against a partial sum for the series. There's software out there for this kind of trickery.

>>12130804
How many papers did YOU publish by the end of your PhD?

>>12131531
1253. Most of it is dragon erotica.

>>12131355
is it true that anime posters are pedos?

>>12131549
Why the pedophobia?

Brainlet here. Can I go about defining the Fourier transform roughly as follows?
Consider a function [math] f:\mathbb{R}\to\mathbb{R} [/math] that is absolutely integrable over [math] \mathbb{R}. [/math]
Extend the function to the two point compactification on [math] \mathbb{R} [/math] containing [math] -\infty [/math] and [math] \infty, [/math] taking [math] f(-\infty)=f(\infty)=0. [/math]
Then we note that this set of functions forms an inner product space, (and a Hilbert space?), and show that the set of [math] e^{i\omega t},\,\omega\in\mathbb{R}, [/math] with [math]e^{-i\omega\infty}=e^{i\omega\infty}=0 [/math] for all [math] \omega [/math] forms a (complete?) orthogonal basis, since over any period, for a nonzero [math] \alpha, [/math] we have [math] \int e^{i\alpha t}\, dt=0, [/math] and we can add countably many of these integrals over a period (all of which are zero) to get the integral over [math] \mathbb{R}\cup\{-\infty,\infty\}, [/math] which is hence zero.
It would then just come down to showing that [math] f [/math] can be represented by an uncountably infinite sum (integral?) over all [math] \omega [/math] of the [math] e^{i\omega t}, [/math] which I imagine would be fairly similar to what would be done in showing that a Fourier series exists for some function.
Finally of course, we restrict [math] f [/math] to [/math] \mathbb{R}. [/math]

Am I close to being on the right track here? Is completeness as mentioned above at all relevant here (do I need more restrictions on the function [math] f [/math])?

>>12131655
None of the functions in the set of exponentials you're proposing is absolutely integrable, thus a fortiori cannot be form a basis.

>>12131370
thanks

>>12131715
Oh, yeah - stupid me. Thanks.

How do I solve the PDE
gyUx - gxUx = 0

Where gy and gx are partial derivatives of some function g(x, y). I'm trying to solve for a U. I understand how to solve
aUx-bUy=0 where a and b are constants, but not the above question with gx and gy.

>>12131754
put it into wolfram alpha

>>12131754
I mean
gyUx - gxUy =0

>>12131769
This kind of post is why I know I'll always have a job.

>>12130811
based

>>12131408
Several Complex Variables

>>12130811
second for based Dempsey

>>12131769
This.
>It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.
>Gottfried Leibniz

>>12130860
>>12130939
>>12130954
>>12130957
>>12130973
One of my mates found a counterexample, I'll write it down if anybody is interested.

>>12131836
>understanding how to do math is tedious calculation

Reminder that if you're EVER going to make it in maths you need to read AT LEAST 1 textbook a week. Naturally, it includes doing ALL of the exercises.

>>12131841
I'm interested in seeing what mistake did he make, post it.

>>12131841
Lmao he must be retarded. We already proved that it's true. How can there be a counterexample?

>>12131891
And be younger than 24.

>>12131892
>>12131896
fuck, the first response is a misclick, I'm talking about>>12130912

bros... i love you... but i love maths more...

What's the minimum mathematical knowledge required to post on /mg/?

>>12131913
post some maths you love

>>12131891
I've never read a textbook in my life. Boring!

>>12131930
Then how do you learn maths?

>>12131911
A counterexample was already posted.

>>12131921
Absolutely none, which should be abundantly clear.

>>12131933
YouTube videos and stackexchange for all homework problems.

>>12131934
Where?

>>12131936
Sooner or later you're gonna have to start reading books if you want to achieve anything of worth.

Is there a more based and so universally applicable mathematical object whose existence is nontrivial to prove yet elementary, than the signature homomorphism
sgn: S_n -> {+-1}?
It's my favorite mathematical object of all time.

>>12130860
This is true, however if you assume only that the curve must be almost everywhere differentiable, it doesn't. Just use the Peano curve.

Math is way too broad for it to be confined to one general.

>>12131933
With my brain

>>12131948
source of the video?

>>12131941
I'll just get some tryhard nerd to do it for me lol. I understand the big ideas, not all that calculation stuff.

>>12130804
Is model theory based?

>>12131966
Parts of universal algebra are finitely based.

>>12131972
Finites can be very big bro

>>12130804
What is the most busted, broken, OP proof technique you know, /mg/?

>>12131891
>all the exercises
How do I do this when some exercises take me like a few hours

>>12131847
>solving PDEs is math

>>12131988
>How do I do this when some exercises take me like a few hours
Didn't say it would be easy....

>>12131988
Proof by contradiction.

>>12131957
Are you sure?
It's not so simple to get the derivative of the limit of a sequence...

>>12131988
>What is the most broken, busted proof technique you know /mg/?

Without a shadow of a doubt:

>[eqn]|x - y| = |x - y + z - z| \leq |x - z| + |z - y|[/eqn]

>>12131988
Exahaustion is horseshit.
>no, no, this isn't true because of this or that intuitive reason, this is true because we've checked a all of the individual cases and they all have particular reasons for making this work

Daily reminder that if you're not voting for Biden then you ain't black!

>>12131938
See >>12131370
it's pretty simple I don't see why you guys think there's no counter examples.

Still yet to be refuted.
https://www.youtube.com/watch?v=LSWIFXP2r14

>>12132141
You don't even understand what the Dedekind definition entails.
You didn't refute it. Your argument makes 0 sense.

>>12131655
Let [math]\mathcal{S}(\mathbb{R})[/math] denote the vector space of all smooth functions [math]f: \mathbb{R} \to \mathbb{C}[/math] such that [math]\sup \left \{ \left \vert x^l \frac{\mathrm{d}f^k}{\mathrm{d}x^k}(x) \right \vert \mid x \in \mathbb{R} \right \} < \infty[/math] for all [math]l,k\in\mathbb{N}[/math], i.e. it's the space of all smooth functions whose derivatives of any order decay super-polynomially fast. This is known as the Schwartz space. On this space one may define the Fourier transform as the linear operator [math]\mathcal{F}: \mathcal{S}(\mathbb{R}) \to \mathcal{S}(\mathbb{R})[/math] satisfying [math]\mathcal{F}(f)(\xi) = \int_\mathbb{R} f(x)\mathrm{e}^{-2 \pi i x \xi}\mathrm{d}x[/math].

Now let [math]L^2 (\mathbb{R})[/math] denote the completion of the [math]\mathcal{S}(\mathbb{R})[/math] w.r.t. the [math]L^2[/math] metric. This coincides with the more common definition for [math]L^2(\mathbb{R})[/math] as the space of all measurable functions [math]f: \mathbb{R} \to \mathbb{C}[/math] s.t. [math]\left \vert f \right \vert^2[/math] is integrable. To be precise, one should take the quotient space where [math]f \sim g[/math] iff [math]\mu(f \neq g) = 0[/math]. Now, [math]\mathcal{S}(\mathbb{R})[/math] being dense in [math]L^2(\mathbb{R})[/math], let [math]f \in L^2(\mathbb{R})[/math] and choose any sequence [math]\{f_n\}_ {n\in\mathbb{N}} \subseteq \mathcal{S}(\mathbb{R})[/math] which converges to [math]f[/math] in the [math]L^2[/math] sense. Define [math]\mathcal{F}(f) = \lim _{n \to \infty} \mathcal{F}(f_n)[/math]. One may show this operator is independent of the choice of representatives and thus we have defined the Fourier transform on all of [math]L^2(\mathbb{R})[/math]. Note however the important, delicate point: this does not imply that for all [math]f\in L^2(\mathbb{R})[/math] and [math]\xi \in \mathbb{R}[/math], the formal integral [math]\int_\mathbb{R} f(x)\mathrm{e}^{-2 \pi i x \xi}\mathrm{d}x[/math] exists!

>>12132156
Not an argument.

>>12132165
Time is like a flat circle....

>>12132141
Schizo captchas when

>>12130811
based and dare I say epic

>>12130811
>>12131784
>>12132282
What's wrong with Nikolaj? Is it the occasional video shilling (good videos btw)? The logic and set theory? How he usually actually discusses math? The 3D women? The Bildschmirfoto filenames?

>>12132341
I don't understand either. He seems cool to me.

How do we know Dedekind's construction of the reals is equivalent to Cantor's construction of the reals?
Specifically, how do we know they're constructing the same exact set?

>>12132866
finding an explicit isomorphism should be easy

>>12132875
But as I understand it, and pardon me if I'm wrong, isomorphisms don't imply an exact equivalence.
For example the set of complex numbers in the form [math](a,0)[/math] are isomorphic to the reals but they are clearly not THE reals.

>>12132885
they are clearly the reals

>>12132885
what are THE reals?

>>12132893
But [math](a,0) \in \mathbb{R}^2[/math] and [math]\mathbb{R}^2 \new \mathbb{R}[/math]

>>12132894
As I understand it, the set of all equivalence classes of cauchy sequences

>>12132899
and what are THE complex numbers?

>>12132157
Thanks!

>>12131372
sphere with 2 points identified is homotopy equivalent to a wedge sum of a circle and a sphere moron

>>12132885
the meaning of isomorphism is roughly "exact equivalence within a certain category"

>>12131754
>>12131774
U = g obviously

>>12132909
[math]\mathbb{R}[i][\math]

How long have you been stuck trying to solve an exercise?
I'm tempted to look for an answer but that would be cheating.

>>12132885
It's not enough that there is an isomorphism, there also has to be an isomorphism which preserves addition multiplication and order. The only axioms a set needs to satisfy to be considered "equivalent" to the real numbers is the field axioms, the axioms corresponding to a total order, and the completeness axiom

>>12133116
After a month it feels like a few millenia.

>>12132885
Just in case you mean this at face value, different models of the reals don't have to be the same sets, in the sense of equality provable via set extensionality. Say rationals [math]{\mathbb Q}[/math] are an equivalence class of pairs of naturals.
If Dedekind cuts are subsets of [math]r\subset {\mathbb Q}[/math] while Cauchy reals are equivalence classes of rationl valued sequences [math]s\subset {\mathbb Q}^{\mathbb N}[/math], how would they be the same.

They are indeed significantly different: From a constructive angle, Cauchy reals without explicit modulus of convergence are not provably Cauchy complete set (all limits being in it). And while Dedekind reals are Cauchy complete, even axioms like Strong Collection, Replacement and Exponentiation don't suffice to show that they even form a set.

Dedekind reals are just the decisions on [math]{\mathbb Q}[/math] that respect its order (as in, the Dedekind reals sort of are reified predicates themselves). So they are complicated to comprehend (I mean this formally, not colloquially).
I've never cared for that bijection, but I think given a Cauchy real, you should easily be able to come up with a predicate in terms of it. To go from a Dedekind real to a Cauchy real, you define a sequence that zooms in on the cut point, which requires to make a left-right judgement.

Why is this wrong?

>>12133177
Bottom is a closed set.

>>12133177
for one, the set defined by the quadratic polynomial certainly contains all points above 1 and whatever the minimum of the parabola is, it's also part of the set and the bottom point, thus the set will be closed.

>>12133184
thinking another second about it, the quadratic equals (x-1)^2, so y=0 at x=1 is the (closed) bottom

Let X be a finite set and T a topology on X. Observe that every point x of X is contained in a unique smallest open set U_x, which is the intersection of all open sets containing x

Show that {U_x| x in X} is a basis for T

How do I show this? I know that x is in U_x for every x but I'm retarded and can't figure out how to show the second property for it to be a basis.

Does any know of more channels with female voices like this one: https://www.youtube.com/watch?v=WZ2SLXrGsNk&list=PLPhi9dVsEVfam7fonbfMB7dkCYGO__med
too toodoo toodoo too doo too doo goddess.
Better than Socratica?

>>12130811
fpbp

What are the maximal ideals of [math]k[x_1,...,x_n][/math] for [math]k[/math] any field?

>>12130811
Criminally underrated post.

>>12130813
Does your shit have a group structure on it tho?

Hate statistics, lads.

>>12133301
Learn probability theory then.

Why does solve returns an empty array? Wrong syntax? simplify works correctly except it doesn't know that x != -3, which mathematica warns about.

>>12133424
ah never mind, it is an expression, should use Eq to make it an equation.

Is there any symbolic algebra package that can also print "show your work" of simplification of fractions, not just giving the final answer? I am looking for something for my pre calculus class to help me solve those tedious and stupid factoring / simplification problems.

>>12133313
combinatorics*

>>12133189
Let [math]x,y,z \in X [/math] and suppose [math] z \in U_x \cap U_y [/math]. According to https://en.wikipedia.org/wiki/Base_(topology) you need to show some [math] \exists S \in T \, s.t. \, z \in S \subset U_x \cap U_y [/math]. Clearly [math] S=U_z[/math] is your choice and it satisfies [math] U_z \subset U_x \cap U_y [/math] since otherwise [math] U_z ' = U_z \cap U_x \cap U_y [/math] is a smaller open set than [math]U_z[/math] which contains [math]z[/math].

>>12131774
Assuming Ux also means [math]\partial_x U [/math] then I think >>12132927 is mostly right - [math] U(x,y) = cg(x,y) [/math] for any constant seem to be the only solutions. Here's an argument:

[math] \partial_x g \partial_y U - \partial_y g \partial_x U = 0 [/math]

Let [math] V(x,y) = U(-y,x) [/math] and note that [math] \nabla V = (\partial_y U, -\partial_x U) [/math] is a 90 degree rotation of [math] \nabla U [/math]. The PDE becomes [math] \nabla V \cdot \nabla g = 0 [/math] implying [math] \nabla V [/math] is orthogonal to [math] \nabla g [/math]. But [math] \nabla U [/math] is a 90 degree rotation of [math] \nabla V [/math] so [math] \nabla U [/math] is parallel to [math] \nabla g [/math], i.e. [math] \nabla U = c \nabla g [/math].

How do you know when to prove by contradiction?

>>12133476
>tedious
no you're just lazy and bad. also leave this thread and never come back

>>12133718
proof by contradiction is just proving the contrapositive. its useful practically because many problems are stated as for all...
but contradiction/contrapositive is phrased as there exits....
This is often easier to work with.
Take the infinitude of primes. By contradiction we have 'there exists finitely many/a largest primes'. This is something we can 'get our hands on', so to speak. As you get better with proofs it will be second nature to try to phrase theorems this way to see if they are more tractable.

>>12133137
>Cauchy reals without explicit modulus of convergence are not provably Cauchy complete set (all limits being in it).
Think this is incorrect, the whole point of cauchy sequence construction is that it's trivially cauchy complete. Every single element in the set is a convergent cauchy sequence.

I think the answer lies in the fact that 2 different completions of a metric space form the same resulting metric space, that is, if you assume the euclidian metric for the Dedekind completion, which is not even mentioned.
The problem then lies in proving Dedekind cuts are a completion of Q

>>12131921
>What's the minimum mathematical knowledge required to post on /mg/?
posting a proof of 1+1=2

>>12133718
>How do you know when to prove by contradiction?
>>12133799
when you are such a brainlet that you are filtered by inquisitional logic.

>>12133189
>>Show that {U_x| x in X} is a basis for T
It's by definition of a basis really

What are the best uses for a double major in Applied Mathematics and Linguistics? I'm studying Applied Math right now but I'm taking a Linguistics class right now and it's so interesting. Are there any fields where it's advantageous to have a degree in both?

>>12133920
no

>>12133920
>What are the best uses for a double major in Applied Mathematics and Linguistics? I'm studying Applied Math right now but I'm taking a Linguistics class right now and it's so interesting. Are there any fields where it's advantageous to have a degree in both?
It's like you're trying to work at McDonald's lmao.

Dude look, the most important principle behind human endeavor is the principle of specialization. If you wanna be good at baseball then play a fuck ton of baseball. If you wanna be a good mathematician then do a lot of math. It's a simple as that.
If you start branching out into completely unrelated fields then the carry over is minimal at best in any direction and you're gonna end up a mediocre linguist and a shit mathematician.
If you want to absolutely branch out, pick a relating field where you know for a fact there's a shit ton of carry over, otherwise you're gonna be running around like a headless chicken.

https://youtu.be/7G4SqIboeig
Turns out this entire thing is actually so obvious it’s surprising it even needs to be said. Literally Aristotle page 2 if p then q of p. If q then z. If P then z, z of p but p not of z. And then you tack some friends along with your ifs. A lot of this is saying look if the if p then q don’t need to be true but it do, what can I say as an if and what would that if then be if it was. The hardest part is getting the question out of your head in the first place because you’re trying to explain something so plainly true it’s hard to even think about so omnipresent is it in your thought

>>12133977
>It's like you're trying to work at McDonald's lmao.

>There are people here who don't have all of the AMS library recommendations

>>12133116
Probably 2 weeks on a particularly rough one.

>>12133718
Eventually you just get a feel for it. The question you should ask yourself is, do I know the right things about what I'm meant to assume? Or do I know more about the opposite of what I'm meant to conclude?

>>12133920
Machine translation and natural language comprehension.

>>12132922
How do you show this?

>>12131408
singular value decomposition
you can aww people because svd can be applied to a lot of modern tech buzzwords like machine learning
my thesis uses svd in attempts to create "fuzzy memory"
basically how much can we compress and can it be recalled and usable
Basically I may not remember perfectly my trip to mcdonalds but I can recall my order and have a good idea on the price

>>12133920
>>12134260
>>12133977
applied math/linguistics can be useful in machine to human linguistics or human cryptography

>>12134260
>>12134317
>>12134327
fake and gay, not math.

>>12134332
Cope.

>>12134339
I'm not even going into math, I just know what is and is not the real mccoy. you are not the real mccoy. no question about it. liars were executed in the court of Darius and they will once again be executed in the court of the coming world-king. Pride is the beginning of the end, anon.

>>12130804
What's an ez-pz point-set topology textbook.

Keep shit easy, I'm a linguist alright.

Pic unrelated.

>>12134474
Bourbaki

>>12134474
Mendelson

>>12134503
>>12134504
I'll check both of these out, but I swear to god if these are tryhard recs I'm coming back

Figured out how to enjoy mathematics more.

>>12134656
glad for you anon :) how did you do it?

>>12134696
Reminder to all trannies

>>12134698
Based.

I was bored. I made this table in Mathematica. (anyone knows how to adjust columns to right?)

>>12133540
Thank You.

/gmmg/

gonna start and finish Limit point compactness from Munkres today and all the exercises from it ^__^ I'm so exicted

>>12133854
The fact that you only quoted half of the sentence makes this a bit awkward.
Anyway, here's a nice paper
https://www.researchgate.net/publication/220082926_On_the_Cauchy_Completeness_of_the_Constructive_Cauchy_Reals

>Every single element in the set is a convergent cauchy sequence.
One note: it's also about defining limit of a sequence of Cauchy reals (i.e. you step through equivalence classes of Cauchy sequences)

y = mx + b

Not exactly math related but I have had the following problem for a while, LaTeX in Wikipedia doesn't load on my phone with chrome(pic rel). It loads on other sites and it loads on firefox (but firefox sucks on Android). Anyone know how to fix this?

what's the point of groupoids?

>>12134925
Maybe using the browser version of the site could help. I can't say for sure.

>>12134949
Consider the homotopy classes of paths in a space, any end points. That gives you a groupoid which has quite a few points.

bros... i want to have sex with all of you...

>>12135219
I imagine having sex with a passive pimply greasy skeleton. They'd not even be good subs if they wanted to, nor would they touch anybody else on their own accord.

>>12131921
You can ask anything as long as you're not stupid with your answers nor you are arrogant 'bout the things you don't know

>>12131936
>YouTube videos
I dare you keep trying to learn homotopy theory through YT videos

>>12131988
>What is the most busted, broken, OP proof technique you know, /mg/?
See [Artin1997]

>>12133177
Third and last one are wrong

>>12134474
Munkres is the most easy topology book out there. Real chads read Engelking's book first

>>12134777
Stop using mathematica and learn a real non-bloated programming language

>>12134777
Imagine having to search for some 3rd party "plugin" just to be able to do what printf does out of the box. And it will probably be full of bugs and shortcomings. Mathematica sucks. Use a real language. Or try combining StringForm with ToString if you have too much time on your hands. But seriously learn Python and use SymPy.

>>12134698
This thread will cease to exist then.

>>12135258
>I dare you keep trying to learn homotopy theory through YT videos
But that's how everyone I know learned homotopy theory and QFT.

>>12135336
This except use Sage.

>>12135258
>Stop using mathematica and learn a real non-bloated programming language
1. Is it really bloated code?

palindromicPrimesBase2 = Select[Range[10000], PrimeQ[#] && PalindromeQ[IntegerDigits[#, 2]] &]
MatrixForm[{#, BaseForm[#, 2]} & /@ palindromicPrimesBase2, TableAlignments -> Right]

2. I know a few programming languages. (FORTRAN, MATLAB, R, C, Pascal/Ada/SPARK...) I'm pretty certain that Wolfram Language is the best tool to make a table of base 2 palindromic primes.
>>12135336
Nice try fromatting strings of unspecified length to the right with printf.

>>12133904
proof by definition

>>12135402
>Nice try fromatting strings of unspecified length to the right with printf.
lol, what? right alignment is the default %20s. left is %-20s, ...
where 20 is some arbitrary max number
thats all there is to that.

>>12131921
There is none. There are some elitist trannies here who think that if you talk about anything other than their favorite math topics you don't belong here since they think it is "their thread". Fuck them. Talk about anything you want, I will reply to you.

>>12134777
Hmmm. Apart from 3 all palindromes in base 2 have odd length. Is it a general property or just coincidence?

>>12135473
Palindromic numbers in base 2 with an even number of digits are divisible by 3. (Also, palindromic numbers in base 10 with an even number of digits are divisible by 11.)

>>12135499
That would be, palindromic numbers in base b are divisible by b+1, wouldn't it?

>>12135410
filtered af

>>12135532
*palindromic numbers with an even number of digits in base b.

Pls respond
>>12115337 #

i still hate set theory

show a single flaw in this guide.

protip: you can't

>>12135696
>Set theory
Well, that was easy.

>>12135696
>principia mathematica
>not Supplementum Geometriae Dimensoriae

>>12135696
The Second Book on Geometry doesn't actually have geometry in the name.

Are differential forms worth checking out?

>>12133268
if [math]k [/math] is algebraically closed, then it is
[math] (x_1 - a_1, ..., x_n -a_n) [/math] with [math] a_i \in k[/math]

>>12135834
Definitely, it's the right framework for multivariable integration, especially divergence theorem etc.

>>12135834
Yes.

>>12135696
Wildberger madness but no actual trig.

>>12135878

Post kino functions.
https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_function

>>12135907
Kek

>>12135977
https://en.m.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification

What's the deal with King Dedede cuts? How do we know there's necessarily one irrational between them, why can't there be multiple? How do we know one can be even made for every irrational.. isn't it circular? Me confume :{

>>12136056
rationals are dense in the seals
any two reals 'next to each other' x,y are distinct
so we can find a rational in the interval [x.y] as y-x > 0
i.e a terminating decimal(k/10^n some k,n a rational) necessary lies between x and y

the bigger question is why |R| =/= 2|Q|

What's the name for the definition of integrals used by Apostol on the first chapters? I remember integrals being defined through limits, but Apostol uses step functions and "ordinate sets".

>>12136072
It's really just Riemann integration but explained in a more primitive way

>>12136056
It was defined to be this way and it hasn’t been shown to not be this way thus it is this way

Is [math]\infty - \infty[/math] ever defined?

>>12136126
It’s basically jailbait, illegal but overlooked when the result is pretty

>>12136126
Let's say you could treat infinity as a fixed number x. Then that would be x-x=0, but for any real number r>0, you have rx=x, and so it would also be x-x=x-2x=-x, and so on.

Just realized that I spent 4 years studying this bullish called mechanical engineering and learned nothing of value. Can you mathchads recommend me a way to get my knowledge on the level of good math undergrad? I want to get into Phd eventually, probably in statistics. Is pic related enough?

>>12136239
This is nonsense and shit.
Simply go to your local uni and take an analysis course.
What I propose is feasible and any level.
I know a dude that was like 50yo family men with full job and responsibilities, became interested in math and started with an analysis class in my uni, kept going till the end and is now a mathematician, purely for hobby.
Dude was an economics major that hadn't done a single calculation for over 30 years, it was hard for him at first but he pulled thru.

>>12136239
if you're over 20, rope

>>12136239
Here's a guide for starting out.

>>12136239
>following the advice of a screenshotted shitpost from 2011
>literally not even a funny meme chart
>seriously recommends buying books and calculates their prices for you
>half of it is all caps
>can't even capitalize properly when not doing all caps
>didn't even get any (You)s, probably screenshotted his own post and reposted it dozens of times
Anon, you're wrong in the head for contemplating seriously following that for more than a second.
Either go to the wiki or go to /sqt/ and eenie meenie a meme chart.
Alternatively, look for advice outside of 4chan.

>>12136360
Quick rundown on pseudoholomorphic curves?

>>12136393
>go to the wiki
Anon... this screenshot is from the wiki

>>12136401
The wiki is that bad?
Never mind, forget 4chan. Get advice from real people.

>>12136420
As opposed to fake people?

>>12136056
>How do we know there's necessarily one irrational between them,
>between
There's no such thing as between any two given rational numbers, there's an infinite amount of rationals between any two of them.
>is it circular
No it's definitional, that's how math works
It is convenient to have a set that has the property that every subset that is bounded from above has a least upper bound, dedekind cuts are just one of the most natural ways to construct a set with that property from the rationals

>>12136239
Why are you posting some meme screenshot and pretending to ask a serious question?

>>12136603
>meme

Having some fun with Heyting algebras.
Here's the the model of negation in a ring for the 3 element one (with False [math] < [/math] Middle [math] \le [/math] True, although the polynomial just needs False [math]\neq[/math] Middle)

[math] \neg(x) := F + (T-x) \dfrac{M-x}{M-F}[/math]

E.g. for [math]F=0, T=1[/math],

[math] \neg(x) := (1-x)\cdot (1-x\,/\,M)[/math]

Of course this has a natural map into the standard Boolean bits representation, as [math]M=T[/math] makes for [math] \neg(x) := (1-x)[/math].

The curious thing for me is that in general Heyting semantics (including Boolean algebra, but trivially), it's always the case that [math]\neg \circ \neg [/math] is monotone upwards.
So [math] \neg \neg P [/math] is, "truer", or easier to prove than [math] P [/math].
And I think the collapse to the Boolean situation must remain an option and [math] \neg \neg P \lor \neg P [/math] is provably out of the question.

>>12136756
([math] \neg \neg P \land \neg P [/math] is out of the question, I mean)

Btw. I dare anyone to find the smallest polynomials implementing the binary operations
We know that
[math] x\land y := x\cdot y [/math]
in the binary representations ([math] F=0, T=1 [/math]). So what's [math] \land [/math] if we add a third value [math] M [/math] with values as in the grid in pic related?
Since it says [math] (0\land x) = (x\land 0) = 0 [/math], I think it's save to make the ansatz

[math] x \land y = x\cdot y\cdot p(x,y) [/math]

and go from there..

>>12134474
https://pi.math.cornell.edu/~hatcher/Other/topologybooks.pdf
hi this is a good list of topology books for every level. depending on what you're doing the "introductory level" may not be useful for you
>>12134656
teach me anon
>>12134698
based
>>12135696
LOL hehe

Is there any difference between calculus and analysis other than the level of rigour?

>>12137025
analysis is a broader term and calculus is a vaguer term

>>12135696
C* algebra? What does marine biology have to do with math!?

XD

>>12136842
Just use polynomials with coefficients in the field with three elements and solve for what the coefficients should be. This is an elementary linear algebra exercise.

I want to learn about knots. Where would be a good place to start for a 2nd year physics undergrad with knowledge of up to Calc 3 ?

I've found a couple of high-end books on the topic which is to be expected given its subject matter.

If anyone can spoon feed me which areas of maths specifically one would need competence in to start an introduction in knots and links that would be nice.

>>12137025
Yes, calculus usually implies applied and is a lot smaller in scope than analysis.

>>12137076
Yes.

>>12137046
xD

>>12137079
Adams, The Knot Book

>>12137079
"The Knot Book" by Colin Adams.
I suggest having some familiarity with the general undergrad topology (up to the fundamental group) & abstract algebra
also this is probably too advanced for you but this:
http://homepages.math.uic.edu/~kauffman/569.html
is a great resource for knot theory.

let [math]w: G\to H[/math] be a group homomorphism. Let [math]k: HSet \to GSet[/math] be a functor such that [math] (M, a) \to (M,b) [/math] where [math] b(g, a) = a (w(g), a) [/math] such that a,b are group actions. Is there a left-adjoint for [math] w[/math] ?

>has boundary points
>is unbounded

>>12130804
>Dude I know wanted to do a math phd
>Went to prestigious university right out of undergrad
>He's now travelling the world going to conferences and getting flown in places by famous mathematicians to do work with them
>Already has like 5 published papers
> he's only on his second year

Is this normal for most math phds or is he exceptional?

>>12137645
The latter.

>>12137645
Either complete larp or you knew it was exceptional and you just wanted to brag on an anonymous image board

>>12137648
>>12137672
No larp dude. Would post his papers but that would mean basically doxxing myself.
I suspected he was exceptional but I didn't know. I honestly don't know how the grad scene is with math.

>>12137645
it's normal in the sense that many people who end up being career mathematicians have a similar experience (although i don't think i have ever heard of phd students "getting flown in by famous mathematicians to do work with them," seems especially unlikely considering the global circumstances)
>>12137672
not certain that this guy is one of them, but there seems to be a lot of people posting intentionally demoralizing/defeatist things in these threads. i don't know if the intention is to troll or maybe they think they're reducing competition for themselves in some marginal way (lol) or maybe they're just bitter meanies. who knows
>>12137677
>that would mean doxxing myself
how would that dox you at all lmao

>>12137677
Ok it's different if you're not talking about yourself then

>>12137704
>(although i don't think i have ever heard of phd students "getting flown in by famous mathematicians to do work with them," seems especially unlikely considering the global circumstances)
Yeah obviously not this year but he has previously.

>not certain that this guy is one of them, but there seems to be a lot of people posting intentionally demoralizing/defeatist things in these threads. i don't know if the intention is to troll or maybe they think they're reducing competition for themselves in some marginal way (lol) or maybe they're just bitter meanies. who knows

Hey man, I don't see how what I posted is demoralizing, it's definitely not my intention to demoralize or demotivate anyone. I'm just curious.
You see, I come from a shitty backwater third world mess. I also want to go into grad school after I'm done with undergrad here. Seeing this dude go to first world, coming from my same uni and everything, and having an amazing time with his phd made me extremely hopeful for my future, that's why I wanted to know if it was normal.

>How would that dox you at all
Trust me dude it would. I'm gonna try crop part of one or something if you're interested

>>12136393
Buying books isn't that bad
(You)s aren't showing
>>12136401
It's not tho

>>12136401
It isn’t, go look at the wiki. Most people on this board aren’t aware of what texts are recommended on the wiki.

>>12136353
Posting in "poutine" or "hon hon hon" is a bannable offense, save that for /int/

Does he unironically have autism? I feel like you have to be somewhat on the spectrum to do multiple videos of this each day.
https://www.youtube.com/watch?v=4ulzbICdw5Q

>>12137831
Yes. Look at his hand movements, saccade, idiosyncratic speech patterns, facial features. He is autistic (based).

>>12137831
No. He is well built. Autistic people do not work out.

>>12137645
>>Is this normal for most math phds or is he exceptional?
what's the topic of research

>>12137887
Logic and topology

>>12137857
>autistic people don't work out
WRONG!

You have no idea how wrong you are it's actually making me smile from EAR TO EAR
>well built
He obsessively rock climbs, an extremely autistic hobby.

>>12137928
>He obsessively rock climbs
Does he actually?

>>12137928
big agree. the strongest guy at my university gym is a math phd turbosperg, and nearly all of the regulars are some flavor of autist.

Any interest in algorithms?

>>12135616
https://arxiv.org/pdf/0905.1680.pdf

Working out is probably one of the most beneficial things for autists to do. Stimming is a big thing, repetitive, easy to see progress: a number just goes up

The deeper I get into topology the more I feel like I should just become a risk manager.

>>12138268
How deep are you?

>>12131948
i feel like that rat is smarter than most humans.

Are all math majors required to study formal logic?

Opinions on Differential Algebra?

>>12138331
No not really. Most just learn Naive Set Theory and that's about it unless they elect to take a class on it.

>>12138331
No, but it's not a bad idea to provided your uni offers it.

>>12138285
Just about to get to Urysohn's lemma

>>12138379
>Drowns

>>12138349
Will it make proofs more straightfoward?

Is libgen also not working for anyone else?

>>12138331
>Are all math majors required to study formal logic?
if they were, most maths would not rely on LEM, so the number of theorems would be 50% smaller

>>12138430
works here, so check dns

>>12138430
Just use
gen.lib.rus.ec

>>12138441
What's LEM?

why do sciences love latex when word equation + word itself are 1000x better and you can save as a pdf to keep formatting

>>12138446
It said "failed server problem" when I tried it.

>>12138448
>he doesn't know

>>12138451
Just use b-ok.org instead

>>12138447
Lie of Excluded Middle

>>12137942
Yes, see his Q&A video, he also talks about his PhD
>>12138256
Das rite

>>12138441
>so the number of theorems would be 50% smaller
I don't concur, you also get a lot more theorems that are classically invisible

E.g. involving
https://en.wikipedia.org/wiki/Subcountability

>>12138613
To add to this, every classical theorem can be faithfully translated into an intuitionistic one, by Godel-Gentzen for example.
So inasmuch as it makes sense to speak of the "number" of theorems, it would increase if anything.

number theorylet here, how do I show [math]xm+yn=1 \implies \gcd(m,n)=1[/math]?
clearly 1 divides m and n, and if [math]k|n[/math] and [math]k|m[/math], then we need [math]k|1 \iff 1 = qk[/math], which only seems to happen if q=k=1, so [math]\gcd(m,n)=1[/math]

>>12138430
libgen.rs works for me

>>12137763
People here do and they hated it last time it was mentioned

>>12138688
[math]gcd(m, n) \text{ divides } xm + yn \text{ for any } x, y \in \mathbb{N} \text{ dumbass, >>>/sci/sqt }[/math]

>>12138719
>linking or quoting breaks Latex
Didn't know about this one.

>>12138693
Tick tock setlets... Typechads are here

>>12138724
literally what did you expect?

>>12138731
I thought the link would just show up in black.

>>12138688
If you're going to be all formal about it, you ought to explicitly state as well that all the variables involved are integers, since allowing rationals would lead to counterexamples like [math]\frac{1}{8}4 + \frac{1}{4}2 = 1[/math].
This isn't just a courtesy to the reader, it's also helpful to you in providing guidance towards the solution: in this case, you can make use of the following nice property of integers:
if [math]kk'=1[/math] for integers [math]k, k'[/math], then this can only happen if both are +1 or -1.

>>12138719
yes i know that. the exercise is showing gcd(m,n)=1 iff 1 = xm+yn though. clearly if i suppose the gcd(m,n)=1, then the result follows.

>>12138784
>needs to show an equivalence
>"how do I prove [one way]?"
Imagine being this unfathomably retarded.

>>12138818
yes i proved one way and am working on the other.

>>12136756
All statements [math] \neg P [/math] are already regular, i.e. [math] \neg\neg(\neg P) [/math] being equivalent to [math] \neg P [/math] as demonstrated by

[math] f \mapsto (p \mapsto f(g \mapsto g(p)))\ \ \colon\ \ (((P\to R)\to R)\to Q)\to (P\to Q) [/math]

which, with [math] Q = R = \bot [/math] reduces to
[math] \neg\neg(\neg P) \to (\neg P) [/math]

Is it possible to power through burnout?

>>12138784
So you still need to prove that if [math]xm + yn = 1[/math] then [math]gcd(m,n) = 1[/math] ?
Well, gcd(m,n) divides both m and n so it divides [math]xm+yn[/math].
Therefore it divides 1 and it must itself be 1.

That's what the anon you quoted was saying.

>>12136133
Dubs of truth.

>>12136126
One of them should be with a hat as I explained previously.

>>12130804
Finally, someone else who knows about this book and how good it is.

Is Shaharon Shelah based? Been reading his work recently and it's pretty darn good.

>>12139026
dude is a master

>>12138842
Burnout is a meme. It doesnt exist

>>12139026
I think you gave him more h's than required.

I'm sure he's good, but who understands his work? I'm into a lot of semi-esoteric math, but I don't actually get what interests the Jews and Enlightenment-nostalgic anglos about those higher stratosphere mathematics? I'd be thankful if anybody has a pitch.
I find it hard to talk with people like Asaf on SE, they are kinda defensive and unnecessarily aggressive.

>>12139048
Damn, started reading him cause a good friend of mine has been collaborating with him. Fucking sick.

is F(x) the indefinite integral of f(x) or the integral from the lower bound to x of f(x)

>>12139089
Neither. The primitive of f.

>>12139097
Tick Tock mathlet

>>12139106
This is a consequence, not a definition.

>>12139089
Neither. [math]F(x)[/math] isn't actually a function.
The differential is a map [math]D : C^1 (\Omega) \rightarrow C^0 (\Omega)[/math]. [math]Ker ~ D[/math] is the subspace [math]K ( \Omega) \subset C^{1} (\Omega)[/math] of constant functions. The homomorphism theorem then guarantees that the induced map [math]D : C^{1} (\Omega) / K ( \Omega) \rightarrow C^{0} (\Omega)[/math] is an isomorphism, and has an inverse map [math]\int ~ dx[/math]. Thus, [math]\int f(x) ~ dx = F(x) + C \in C^{1} ( \Omega) / K(\Omega)[/math] is written according to standard quotient notation.

Find

[math] f(n) = A\, f(n+1) + B\, f(n-1) [/math]

for [math] n\in \{0,1,2,3,\dots, N-2, N-1, N \} [/math]

in terms of [math] f(0) [/math] and [math] f(N) [/math].

>>12139190
I hit it with Mathematica and it seems this is somewhat trivial (for it) if B=1-A, but otherwise you get [math] \pm \sqrt{1-4\cdots} [/math] terms.

why 0!=1 but n/0=undefined ?

>>12139300
[math]\Pi (0) = 1[/math]

>>12139430