/sci/ - Science & Math

Lie Algebras

>>12570185
Infinitesimal groups

Is anything defined on a manifold in a coordinate free way a geometric object?

I start my first analysis class in a week (Real). I'm so nervous.

>>12570302
Spam the triangle inequality in every proof. Know how inequalities work. That's literally analysis 1.

>>12570302
Do you know how to add by zero? Multiply by one? If so, you'll do fine.

Q: ¿What is it called when two post-op trans women have sex?
A: Cohomosexual, since it's gay but now the arrows are all pointing in the opposite direction.

>>12570558
yikes

>>12570302
square or multiply by conjugate/conjugate
thats about half of the course

Do you guys understand how this is derived?

>>12570558
please leave

>>12570175
I don’t completely understand the proof in example 20. How does this map to the positive integers? He says we arrange them to arrange them we start with p/q such that p+q = 2, but how does this exactly work? For example if we had f(1) = 1/1, f(2) = 1/2, f(3) = 2/1, doesn’t this fall apart immediately? F(4) = 3/1, but how about f(5)? We can skip 2/2 since it was already listed, but when we reach 1/3, that p+q doesnt = 5, it equals 4 again. So how does this work as a one-to-one correspondence? Doesn’t this violate f(n) = f(m) -> n = m? 3/1 and 1/3 both add to 4.

>>12570723
Sorry, forgot the image.

>>12570175
>completed infinity
Is this the biggest brainlet filter?

Why are the minimal fields of definition of a projective point isomorphic/equal? Or better, how can I show that they are equal/isomorphic to the subfield of the algebraic closure fixed by the Galois subgroup of automorphisms fixing that point?

Define the compact-open topology of [math] C(X,Y)=Y^X, [/math] continuous functions from [math] X [/math] to [math] Y, [/math] to be generated by the subbase of sets [math] [K,U]=\{f\in Y^X:f(K)\subset U\} [/math] where [math] K\subset X [/math] is compact and [math] U [/math] is open in [math] Y. [/math]

Show that if [math] X [/math] is locally compact, then [math] Y^{X\times Z}=C(X\times Z,Y) [/math] is homeomorphic to [math] (Y^{X})^Z=C(Z,C(X,Y)). [/math]

This good?

>>12570302
undergrad level analysis is just about learning a few tricks. conceptually the material is very easy. do every exercise in baby rudin and you'll be fine.

is this allowed?

[eqn]f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/eqn]

[eqn]f'(x) g(x) = \lim_{h \to 0} \frac{f(x+h)g(x) - f(x)g(x)}{h}[/eqn]

>>12570292
Tangent space at p as (m/m^2)* where m = {f | f(p) = 0} and m^2 is every product of two functions in m.
A tangent vector is basically a direction with respect to which you can derive a real valued function, and the derivative is a linear functional defined natural on m/m^2.
I view it as
1) you consider functions that are zero in p because for every g(x) non-zero at p you can correct it by a constant, obtaining g(x)-g(p), and then you have a function in m whose derivative is the same as g(x)
2) Leibniz rule translates to "if h = f*g and f(p)=g(p)=0 then h'(p)=0" so m^2 gets sent to 0 by the derivative.

>>12570292
>>12571685
Reread the question and it wasn't an answer, my bad.

>>12571682
This follows from a very basic property of limits, you should know that.

>>12571721
could you be more helpful than this though

>>12571750
just look at it
please don't shit up /mg/ with questions that can be solved by looking at definitions

>>12571750
What have you tried?

>>12571721
g(x) (and f(x)) are constants for all you care in that limit. h is your variable there, not x.

>>12570558
That was funny. I like you.

>>12570637
in the paragraph above you have that [eqn]\frac{p_{2k}}{q_{2k}}<\alpha<\frac{p_{2k+1}}{q_{2k+1}}[/eqn] and
[eqn]d_{2k}=q_{2k}\alpha-p_{2k}[/eqn]
[eqn]d_{2k}=p_{2k+1}\alpha-q_{2k+1}[/eqn]
and also note that
[eqn]0<d_n<\frac{1}{q_{n+1}}[/eqn]
To get your iequality us [math]d_n[/math] to show that [eqn]0<k\alpha-\frac{kp_n}{q_n}<1[/eqn]

>>12570558
I kek'd and then re-evaluated my entire sense of humour

>>12571468
What have you tried?

>>12571758
>please don't shit up /mg/ with questions that can be solved by looking at definitions
I love how this is considered to be shitting up but the reddit-tier trans joke got no snarky reply from you
How about you consider fucking off this thread completely? /mg/ welcomes everyone and if it helps them appreciate math more, let them ask any questions, from calculus to K-theory.

>>12571682
What happens with g(x) when you approach the limit? Pay more attention to what is actually changing.

>>12572080
Let's see... let's say our point [math]P[/math] is in both ith and jth charts, then the bijection [math]\varphi_j^{-1}\circ \varphi_i |_{\{P_i\}} :\{P_i\} \to \{P_j\}[/math], where [math]P_\ell := \varphi_\ell^{-1}, \ \forall \ell\in \{0, 1, \ldots, n\}[/math] induces a coordinate ring isomorphism [math](\varphi_j^{-1}\circ \varphi_i)^* : K[\{P_j\}] \to K[\{P_i\}][/math] which is what I wanted?
Those fields are clearly contained in the fixed field I mentioned in the second question, but I don't know how to proceed in the other direction, looks like Galois theory... I guess I just need to show that it's equal to [math]K(P_{\mathbb{A}^{n+1}})[/math] (image of [math]P[/math] in [math]\mathbb{A}^{n+1}[/math]). The Galois subgroup fixing this field is the same, so by Galois correspondence the fields are equal? The extension is assumed to be separable and is obviously algebraic, so it's Galois.

>>12572085
The whole general is shit, but you can hardly give 295/300 replies in each general a (You) to tell them to stop posting.
>/mg/ welcomes everyone
Oh well fucking well done on that. This general has driven away pretty much every decent poster it's ever had by now, and there's fuck all actual discussion of any substance. These threads are just an eternal summer, 95% questions that can be answered by literally reading basic definitions in books, or "do my homework" posts from high school dickheads who should be banned anyway, people chatting about nothing at all, the occasional Wildberger nigger, and then basically anything else that isn't mathematical. When you can see some discussion taking place it's become surprising nowadays.

>>12571899
I still don't get it...

>>12572225
I agree that he should know enough to know that his question shouldn't be asked, but I don't get why you angry elite posters care so much. It's not like there isn't space for you to carry your conversations in parallel. If it's that what you want, go to math overflow or something.

>>12572225
Bro just post some math then
Post some complex aalsys

>>12571682
Yes its allowed because you get the same result, the g doesnt interact with the limit expression try factoring it

>>12570558
brilliant, thanks

>>12570185
How is this different from inner product spaces? Whats the deal with this?

If you think about it it is kind of weird that we can represent two variables with one object when we use color and brightness for complex graphs.
In reality, color and brightness is not actually viable and contradicts the notion of fitting two objects onto one spatial dimension, because a color or sound is amplitude and frequency, which requires two dimensions. It only works on computers because they use pixels which create planes.

Conjecture: Only one concept can exist at any point in space.

Mathphys gurus, I want to study the so called pseudo-relativistic Schrodinger equation which has an operator of the form [math]H_f:=\sqrt{P^2+m^2}[/math] with [math]P=-i\hbar\nabla[/math] the usual momentum operator. Now the rigorous definition of [math]H_f[/math] I have found uses pseudo-differential operators, as the "inverse fourier transform" of a function with symbol [math]\sqrt{x^2+m^2}[/math]. Now what I believe is that this definition coincides with the definition of such an operator using the unbounded functional calculus of [math]P^2[/math]. I think this, because well the spectral decomposition of the [math]P^2[/math] is literally given by the Fourier transform, and the maximal extension of the operator has a domain of functions such that the multiplication of the symbol with their Fourier transform is in [math]L^2[/math] which I believe is the same domain as the operator given by the functional calculus as pic related. I just want to see if my intuition is right before trying to prove it rigorously or if I'm doomed to fail.

>>12570637
lads?

is the Radon-Nikodym derivative linear? if i have two absolutely continuous signed measures then i'm pretty sure that it's true, but what if i have a signed measure that is decomposed in a ac part and a singular part? How do i compute the derivative?

>ZFC is consistent because we haven't found a contradiction yet
>Are you saying twin prime conjecture, which is an infinitely simpler proposition that the consistency of ZFC, is true because we haven't found a disproof yet? You can't do that!

Doing a-level maths
about half a year into my two year course
>stats is just writing out excel sheets onto paper
>mechanics is physics disguised as math
>pure is trigonometry right now with some algebra stuff sprinkled in
Does it get better?

I have 5 months to learn maths from zero to a freshman level. Is khan academy + Serge Lang acceptable?
What do you think?
Thank you

>>12573173
The AC is the derivative of sine

How does this make mg feel?

>>12573606
Quite confused.

>>12573606
Only retards and autists complain about these sort of arguments. Anyone with basic bitch knowledge about calculus can make this argument rigorous in terms of limits and see it is basically the same thing.

>>12573614
How are you today emgeetranny? It[s awful quiet on here.

>>12573624
Please go ahead ans show us

>>12570637
guys can you help pls

>>12573634
Try working through the notation definitions with pen and paper

>>12573629
The length of a cord of an arc of measure [math]\theta[/math] is [math]2\sin(\theta/2)[/math] and as anyone with basic bitch knowledge of calculus knows you can use Taylors theorem to express this function as [math]2sin(\theta/2)=theta+r(\theta)\theta[/math] where [math]r(\theta)\to 0[/math] as [math]\theta\to 0[/math]. Now express everything through this representation and take the limit at the end and voiala, instead of using approx you just put explicitly the remainder.

>>12573669
Arigato Gozaimas. There really needs to be feynman diagrams for rigor because its too cumbersome currently.

>>12573661
I am...

>>12573625
>It[s awful quiet on here.
Newfags scared the "post real mathematics or gtfo brainlets!!1" elite away, oh well

>>12573625
A bit foggy in my head, thanks for asking. I was thinking about this little setup where I have a nice enough ring [math]R[/math] and some ideal [math]I \leq R[/math], and then I take the canonical projection [math]p\colon R \to R/I =: S[/math]. Now, this then induces a functor [math]p^* \colon S\textbf{-mod} \to R\textbf{-mod}[/math] which turns a module over the quotient ring to a module over the original one via that projection. It has a left adjoint, which in this case will send projectives to projectives, but this right adjoint doesn't in general. However, I had this nice chain complex satisfying some technical conditions (which are not necessarily satisfied by any ring, so I want my ring to be for example von Neumann regular to guarantee its existence, but no one cares about that now). I would take a projective module over the quotient ring, send that to the category of modules over the original one and take the complex. Should the image of my module be projective also in [math]R\textbf{-mod}[/mod], then the complex would by acyclic. I had high hopes that the converse would hold, that is, acyclicity would imply projectivity. However, that seems not to work, quite sad. Consequently, I cut myself... I was getting rid of hair on my arms, and then I coughed and now there's a scratch on my wrist because of that, oops! I'm also in a 10-day self isolation from traveling, but I have literally no idea how many days have past because I didn't put the prison style day markings anywhere. 5th or 6th.

How about you? Have you done anything interesting related to anything? Or anything of any sort of any relation to anything?

Can I get an example of a method of enumeration of all possible polynomials with integer coefficients? The rationals can be enumerated as a line moving back and forth across a 2-dimensional lattice, can this mechanism work for an infinite-dimensional lattice?

>>12573717
The topic that motivates this question is the most boring thing in all of mathematics dude, why do you think abstract algebra was invented? It's to make that shit bearable.

>>12573407
you do some interestnig sequence stuff and possibly polar coords iirc on top of plug n chug polynomial/ln/e/trig product rule and inverse product rule >calculus with some composite function differentiation/integration, similar to first year american uni stuff
any further maths?

>>12573717
https://youtu.be/soHBNEJlzL0

>>12573717
>all possible polynomials
how many variables my dud

>>12570637
Ah shit I finally figured it out.

If p/q is of an even convergent, then you end up with
np/q < na < np/q + 1

We have
[np/q] < np/q < [np/q] + 1 < np/q +1

There's two places to put a: before or after [np/q] + 1
but since a is below 1, it has to be before

Now,
If p/q odd convergent then
na < np / q < na + 1

We have
[na] < na < [na] + 1 < na + 1

Again, two places: before or after [na] + 1.
But since, for odd convergents we have p/q < a, we get that p/q < 1.
So, it has to be before [na] + 1.

>>12573717
Think of a polynomial in one variable as a sequence. Let's say we have [math]f\in R[x][/math] for some ring [math]R[/math]. Then [math]f = (a_n)_{n \ge 0}[/math], where each [math]a_n \in R[/math] is the coefficient of the nth power of our variable. Furthermore, there is a maximal index for non-zero coefficients, so you are essentially talking about the set of those integer sequences that have only a finite number of non-zero entries. I think, yet I'm not completely sure, that you could then take all the primes, and then order them somehow (let's just go with the natural order), and then define a function [math]R[x] \to \mathbb{Q}[/math] by [math](a_n)_n \mapsto \prod p_{n+1}^{a_n}[/math], (n starts from 0), and this should be a well-defined injection. If this was the case, then it would be a countable set. Err... Exercise of the night: prove or disprove my post.

>>12573762
wait nvm... it's na and np/q, not a and p/q

>>12573717
An infinite cartesian product? Isn[t that supposed to be uncountable?

>>12573706
I guess I was thinking about line integrals earlier. Was gonna look up how QED works cause path integrals , thx for reminding me
Was also watching a vid on Neural Nets someone posted on sci, and wondering about human memory

>>12573730

He could get around that with the qualifier "univariate".

>>12573851
>An infinite cartesian product?
With finitely many nonzero terms. Yes, it's countable.

>>12573861
He said all polynomials, infinite slots

>>12573870
Think about it for a little bit.

>>12573717
If you've got a countably infinite list of polynomial variables, say [math]v_1, v_2, v_3, v_4...[/math] how do you even enumerate their various powers?
Obviously [math]{v_1^0, v_2^0, v_3^0, v_4^0...}[/math] can be considered the first element of the list as it corresponds to the constant functions and [math]{v_1^1, v^0_2, v_3^0, v^0_4...}[/math] can be considered as the second element, but which of [math]{v_1^2, v_2^0, v_3^0, v_4^0...}[/math], [math]{v_1^1, v_2^1, v_3^0, v_4^0...}[/math] and [math]{v_1^0, v_2^1, v_3^0, v_4^0...}[/math] is the third element?

>>12573706
>[/mod]
Now that's just embarrassing!

>>12573851
Are you a mathemagician or do you consider yourself more of a physicist? Considering the fact that you also watch a video on neural networks, you seem to be interested in all sorts of things. Are the line integrals in QED path dependent? The classical example of getting a barrel into the back of a car by lifting it or by rolling it up a ramp is an example of work being independent of the path, but I would imagine quantum stuff to be more fragile in that sense. Just a gut instinct, though.

>>12573706
When more pics whore?

>>12573892
He was presumably asking for all polynomials in one variable, for which the answer has already been given.

Now if you want to enumerate all products of monomials, then this is solved by the same enumerations.

But to find all polynomials with integer coefficients with also countably infinitely many possible variables, that another questions - and one probably nobody asked.
That said, it feels like this is also just countable. Since all things being counted here are just finite length polynomials in any case.

Could someone please help me with a sine wave that I could plug into desmos that resembles a graph like pic related?

I just need the irregularity. The position of the spikes and valleys are irrelevant.

>>12573953
>QED
I still dont understand the formalism but it seems like the idea is to take a bunch of path integrals and average them out, and certain ones are more probably so more weighted, idk how thats determined but it likely has to do with geodesics . So that would mean that each path gives a different result
>math or physics
I think both are important, and equal. Math is with psychology and philosophy, more platonic forms, and physics is with the material realm, but theyre matched up

>>12573970
Well a sine wave is not irregular and who knows which special functions the online tool knows of. Why does it have to be Desmos? What do you even need it for?

>>12573970
Discretize a sine wave in excel and add pink noise

>>12573970
best I can do you for on the fly, just uses standard python

https://gist.github.com/Nikolaj-K/8c56fe6f3c0287b34a130652dbf338cd

>>12573963
Never. Besides I'm chubby and ugly atm. I'm sorry for even posting the original one. It was my birthday, my sister took me out for a drink before she went to work, and I had maybe 3 or 4 in a short period and then I thought it would be a good idea...

>>12573975
>I still dont understand the formalism but it seems like the idea is to take a bunch of path integrals and average them out, and certain ones are more probably so more weighted, idk how thats determined but it likely has to do with geodesics . So that would mean that each path gives a different result

>>12573975
Do you already know the methods how you can find those geodesics? It will probably lead to manifolds pretty quickly (if it already hasn't), so a lot of local coordinates and stuff. A quick guess would be that the weight depends on the local electric field's magnitude somehow, and then you would have an integral. Let's say you had a path [math]\gamma \colon I\to M[/math], where [math]M[/math] is a manifold. Then you could take a function [math]w\colon M\to \mathbb{R}[/math] and the integral [math]\int _0^1 w(\gamma(t)) \gamma(t) dt[/math] would be weighted by that maybe... I don't know. Just throwing random ideas around.
>I think both are important, and equal. Math is with psychology and philosophy, more platonic forms, and physics is with the material realm, but theyre matched up
I think this is a good way to look at them. No need for unnecessary fighting.

>>12573983
Doesn't have to be desmos. I made a trig engine for a simple reason: I need to "raycast" through a bunch of data points in an array to compare that value with the ones next to it. And the data points form a wave like the btc prices. Built this tool to test it

Thanks <3

>>12574052
Pretty please :3

>>12574034
Can't seem to install fucking python on w7.

Thanks for the source though, will port

>>12574085
Sorry but no. I look so bad in that pic.

>>12574052
This is weird. It seems like you[re correct, that its just the various integrals over the E field. But as far as I can tell E fields are conservative vector fields. Maybe wavey perturbances fuck with it?

>>12574135
:(

>>12574205
Well, integration is the only sensible way to do it if you want to change your path in some continuous way, I think. Where the dependence comes from, no idea about that. Could it be that [math]x - x_c[/math] can be absolutely awful if your field is turbulent enough? Then those [math]a_n[/math]s could be painful to compute. Or something... I really don't know. Please tell me when you find out.

>>12574135
You really are a useless whore. You don’t even say “no I have a bf” instead you say “no I look bad”. I’m reporting you to your bf.

>>12574251
>tell you when I find out
Will do assuming you[re still posting on mg by then. I dunno how long itll take, sometimes I forget about things for 6 months or 2 years... Or maybe itll click tonight. Haha, life

>>12574135
Post it whore.

Oh I got it, its the action not the E field
So farther away things have worse actions and are valued less in the sum

>>12574314
This is /mg/ please have some self respect

>>12574270
Please don't wake him up. He has some stupid energy business course that starts tomorrow morning and it made him grumpy. But I guess you are right otherwise. A feeling of guilt creeps in. The darkness outside finds a home inside me. I guess I am a horrible person.

>>12574277
Try to do it in a year please. Then I should still be around for sure. But even if I don't see it, please make sure you find out how things work. Don't leave things unfinished.

>>12574314
No. Read >>12574270

>>12573975
A path integral, subject to boundary conditions [math]\Sigma[/math], is a Liouville probability expectation [math]\mathbb{E}_\Sigma[O]\equiv e^{i\Psi_\Sigma[O]} = \int d\mu_\Sigma(\psi,A) O(\psi,A)[/math] over the space [math]\mathcal{D}[/math] of the degrees of freedom, be it connections [math]A[/math] (gauge fields) or particles [math]\psi[/math] (spinors/tensorial fields) or any other dynamical things you may put into your theory. In Euclidean field theory, you're right in that the integration is weighted by the Gibbs measure [math]d\mu_{\psi,A}=d\psi dA \exp -iS_\Sigma[\psi,A][/math], where the Boltzmann factor [math]\exp -S[/math] tells you the probability of a configuration [math]\psi,A[/math] (satisfying boundary conditions [math]\Sigma[/math]) being occupied.
Now the problem is the "Liouville" part: we need the WKB wavefunction [math]\Psi[/math] to describe dynamics, which means that we must put a symplectic or at least a Poisson structure on the moduli space [math]\mathcal{D}/\mathcal{G}[/math], which is usually infinite dimensional and not even a manifold. For general interacting QFTs (i.e. not a TQFT/CFT/susyST) this is still an open problem, because finding the appropriate [math]d\mu[/math] is equivalent to quantizing interacting fields. For free theories [math]S_\Sigma \sim |\mathcal{A}\psi|^2 + |F|^2[/math], however, you are able to leverage the zeta-determinant/heat kernel technique to evaluate the path integral. This involves an expansion in [math]\Sigma[/math]-eigenmodes of [math]\mathcal{A}[/math] in order to decompose [math]d\mu \propto \prod_n d\psi_n da_n[/math]. You can then endow each eigenmode an individual Poisson structure when quantizing.

>>12573975
>>12574052
>>12574205
>>12574251
Path Integrals have really nothing to do with line integrals. Physicists think of path integrals as functional integral analogous to the functional derivative in calculus of variations. Some even mention that it is integration against a measure on an infinite-dimensional space of paths that satisfy some boundary conditions. The best example of these sort of integrals come up in Stochastic processes https://en.wikipedia.org/wiki/Law_(stochastic_processes).. In QM the idea is that if [math]S[/math] is the classical action functional then the probability that a particle in that system moves from a to b is given by [math]\int e^{i\hbar S}Dx[/math]. Sadly, this integral is in general ill defined and there is till today no general theory of path integration that seems to be applicable to all physical systems. Worse, in QED is an integral over field configurations, not paths, which makes it even more difficult to give a precise meaning to the integral.

Now, the idea of things depending of their path in QM does have some merit https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect but you don't need path integrals to derive this result.

>>12574335
I think I understood enough to keep my dumb algebraist fingers away from my keyboard to prevent myself from even trying to write any more dumb ideas. Thanks for the explanation. An integral over field configurations sounds like an abomination. No wonder there is no general theory. Scary stuff.

>>12574334
>Gibbs measure
Are we summing over measured functions? Where does the gibbs measure come from, is it related to the hamiltonian/action?

>>12574362
Yes [math]\mathcal{D}[/math] has a functional and an algebraic component, both related by representation theory. For instance fields [math]\psi\in W \otimes H[/math] have a functional part (lying in some appropriate Sobolev [math]W[/math]; cf. GNS construction) and an algebraic part (lying in some finite-dimensional Hilbert [math]G[/math]-rep space [math]H[/math]; cf. Peter-Weyl theorem). Here [math]G[/math] is your local Lie gauge group.
>Where does the gibbs measure come from
Statistical mechanics.

>>12574388
>statistical mech
What hapens at really small length scales when everything is discretized? Like if an interaction occurs a few plank lengths apart

>>12574052
>I'm chubby
Can someone post the original pic? I was never interested before but a chubby math trap gf sounds pretty cute ngl.

>>12574439

How would we go about finding conserved quantities of a PDE in the complex domain?
Like let's say we have a PDE which has one of its variables made complex, which we might do in a 2D problem. I actually can't find much at all which is written about how to proceed with those types of PDEs.

Question. Is the following true?
[math]\mathbb{Z}_n[/math] is an integral domain if and only if [math]n[/math] is prime.

>>12574669
Yes. Otherwise there is a proper divisor d of n so dp=n, for some p not equal to n.

I need to take a break from math to learn german... God this sucks... Realities are changing... Powering is amassing..... Send me blessings mg...... Brb......

I'm doing some work on degree-3 polynomials in four variables. In notation, if I write x^3 before x^2y which is before xy^2, which is before y^3, and I write y^3 before z^3 before w^3, what's the correct order of terms out of x^2w, xz^2 and yzw?
Is there some kind of manual for how it should be worded?

>>12574669
0 isn't prime and [math]\mathbb{Z}_0 \cong \mathbb{Z}/0 \mathbb{Z} \cong \mathbb{Z}[/math] is an integral domain.

>>12574705
Here comes with autist with the usless answer.

>>12574710
>NOOOOO YOU CAN'T CORRECT ME WHILE I'M SHOWING OFF MY BASIC KNOWLEDGE OF ALGEBRA I
Cope.

i have exams in june and i need to learn from basic algebra to caculus 1? is it possible? and i have other subjects so is it better for me to focus on math only for 2 months or study other subjects with math for 4 months?

>>12574135
I liked it, and I'm not even gay

>>12574439
Well, somebody posted those, sadly.

>>12574669
For n>0, yes. Otherwise you can do that silly n=0 thing, or even take a negative n which wouldn't be prime and still do the same stuff. That's why n>0 is often mentioned.

>>12574695
Choose the ordering for your variables, then write them in that order with decreasing powers. If x comes before the rest, then y, then z and then w, you write [math]x^2y + y^2z + z^2w + xw^2 + yzw[/math] etc.

>>12574750
No you didn't. I'm fat and ugly.

>>12574470
I wonder if you're the girl I saw at a conference online last semester. How many trans women are algebra/topology grad students right now in the US? Can't be more than a couple. If you're down to do queer stuff with a girl for a change hmu, you'll know who I am eventually.

>>12574735
Study everything with math for four. Honestly, if you’re a smart guy, it shouldn’t be an issue to get through Calc I starting from algebra.

>>12574917
Sorry to disappoint you with these things, but:
(1) I've never had my camera on.
(2) I don't know about the situation in the US, as I have never visited the whole continent.
(3) I'm not into girls.
(4) I already have an owner.
Nevertheless, I thank you for the offer and wish you luck in finding someone to do things with.

Good night, /mg/.

>>12574917
She's euro.

>>12575008
Good night and sorry for posting your legs (not really sorry)

>>12575008
Fair enough. Best of luck in your future endeavors too.

>>12574917
You that fag who solved that Conway knot problem like it was nbd?

>>12574753
Negatives of prime numbers are prime.

Proof:

Let p be a positive prime. Then p is not a unit and is nonzero so -p is not a unit nor do we have p=0.
Say -p|ab. Then ab=m(-p), so ab=(-m)p, so p|a or p|b. WOLOG p|a; then a=np, so a=(-n)(-p).
Then -p|a, so if -p|ab then -p|a or -p|b. Since p is a nonzero nonunit then p is prime.

>>12575091
What the fuck lmfoa

>>12575021
To be honest, you have no reason to be sorry. It was my own stupidity to even post them in the first place. A misjudgment of how degenerated some of you are.

>>12575071
Thanks. Are your things progressing well?

>>12575091
Prime yes, prime elements. Prime numbers are greater than 1 by definition, though.

>>12575154
Don’t lie you slut you probably enjoy the fact that some fags ITT coomed to that pic.

>>12575154
>Thanks. Are your things progressing well?
Yeah, I hope to get the two papers I'm working on published this year and then I need to get a job (and as mentioned earlier maybe a gf haha). Thanks for asking. :)

>>12575157
I made no comment on that. I meant the fact that someone asked and someone (same or different) solved a problem I gave as the condition for a mentally ill person's legs that look like those of a sumo wrestler in that pic anyway. If someone gets joy from that pic, that is only a positive thing. It is a rare occasion of me being of any use to anyone.

>>12575179
Nice! Do you have many publications? By a job, do you mean something outside academia? Do you have a good rainbow scene in your city for search purposes?

>>12575091
What is gained by including negative factors? Is there a new property added that makes up for numbers losing their unique prime factor string? I feel like there is something useful giving every prime four factors but I don't see what it is right now. 1 and -1 only have two factors which specifically make them not prime. Is there a way to make i relevant? Maybe 1 isn't prime, but -1 is. Ah, I do think so:
Including i makes all negative numbers not prime.. oh but also all positive numbers are not prime. Actually, 1 and -1 are defined by i and all primes factor to p and some combination of i including pi numbers. A number like 9i also has compositions using each of its factors once: 3, -3, 1, -1, i
Since 1(-1)(i) product is -i, only prime square complementary numbers are candidates for these compositions consisting of exactly 1 of each of its factors. hmm

>>12575190
>Do you have many publications?
Just a few, I'm still a grad student.
>By a job, do you mean something outside academia?
No, I only want academic jobs really.
>Do you have a good rainbow scene in your city for search purposes?
Idk, my love life is crazy. I willing/hoping to relocate to somewhere with more gay people.

faggots fuck off

>>12575281
*I'm willing

Okay it's really time for bed, ladles.

In my third year of math undergrad. I'm at a decent school in the USA and am taking decent classes (last semester, 1st semester graduate Algebra and 1st semester graduate Analysis, which was complex analysis). I get As in math classes, with some A- grades in the more trivial stuff. I want to do math for a living and I will be applying to grad schools in fall of this year; my advisors say I can apply comfortably to good tier graduate schools.

However, I believe the US is rapidly becoming a failed state. I don't know if I will be able to survive in a college town for 5 more years, and I'm worried that there will be no economic opportunities for mathematicians in the USA. I'm considering emigrating to somewhere like Singapore or HK, and maybe even going to graduate school there. Other places I've considered moving are Uruguay and

Any ideas for what I should do about this problem? I can't ask my advisors this since I don't want to look like a paranoid rightist in front of a college prof.

PS I want to study Algebra; I speak excellent spanish and fair Cantonese

>>12575281
>Just a few, I'm still a grad student.
What is it like to publish something?
>No, I only want academic jobs really.
I thought being a student is sort of like a job.
>Idk, my love life is crazy. I willing/hoping to relocate to somewhere with more gay people.
Hopefully you find some nice place for that. Good night!

>>12575102

For doing math in /mg/ you laugh at me? Shame.

>>12575222

Even if we stick to positive prime integers, there still isn't a "unique factor string", since we can permute the order of our factors. 2*3=6=3*2.

However, the list of factors of an integer is unique up to the action of a group, whose transformations consist of permuting the elements of the factor list and then multiplying each one by a unit. In some rings, this fails; frankly, they are more interesting than those rings in which it succeeds.

>>12575368
nice redditspacing

Two actually good episodes of /mg/ in a row?
What the fuck is happening? Does bullying actually work?

>>12575190
Chubby trannies, especially the legs get me off like nothing else, besides maybe dabbing on bourbakists and tarskists brainlets

>>12575309
Important question, is your weewee still there?

>>12575729
Well, I hope BMI ~18 is enough for you.

>>12575738
Yes. I don't like it, but it's the less disgusting alternative. Nick Land was wrong when he wrote that neo-vagina arrives from the future. It won't, yuck.

>>12575828
Nick land is a dummy who went a little hard on amphetamines, and couldn't even bother to write one paper on graph theory. I hope you don't do the same, because it'd ruin the delicious fat distribution you've got going.

>>12575838
I don't do drugs, but I can't write a paper of graph theory either. Thank you for your kind words, even though you self identify as a chubby chaser.

Speaking of graph theory:
>Towards Stratified Space Learning: Linearly Embedded Graphs
https://arxiv.org/pdf/2101.04375.pdf
Abstract non-sense
>Kan extensions are partial colimits
https://arxiv.org/pdf/2101.04531.pdf
For those interested in CFT
>Microscopic Models for Fusion Categories
https://arxiv.org/pdf/2101.04154.pdf

>>12575828
>Well, I hope BMI ~18 is enough for you.
You think that's chubby? I guess it explains your sharp knees in the photo.

>>12575878
I didn't mean it as in calling you fat, anitranny. You've got a feminine distribution of body fat, of which I'm not sure you understand the importance in passing. Also bmi of 18 is the same as what my mom got, with her typical build she's deffo not fat. Remember, if you go anorexic, you'll lose the estrogen from the fat, how much is that in pills?

>>12575885
Yes.

>>12575907
Ah, sorry for misinterpreting you. Fat is a nasty thing in its role as a necessary evil. Anorexia also debilitates one's brain, so that's another reason to try to live with body fat. Thanks for being nice! Are you still on Christmas break or have you started working on stuff already?

you will NEVER be a women

>>12575965
I always work on stuff and have breaks only when I run out of meds

>>12575970
good

>>12575970
>transphobe
>low iq that can't use english correctly
what a surprise

>>12575970
I wish I had your diligence. What are you working on and which meds are we talking about?

>>12575970
I know.

Literally all I've done for the past week is solve problem after problem after problem of 3D distributions in E&M books. I've never heard of such a retarded habit. I literally just pick up Jackson, solve a problem, pick up Jeans, solve a problem, and keep going until it's night.
Is this autism?

>>12576038
Actually, I've really been doing this for the best part of a decade. I've just ramped it up to ~5ish problems a day lately.
I wish this would actually be useful in some way lol

>>12575987
the bad kind (not personality disorder). Work's embarrassing, don't want to talk about it. Maybe I'll shitpost about it later, you'll never know.

>>12576065
Please take care of yourself anyway.
>you'll never know
Maybe I can recognise your style and connect a few dots. Who knows. Anyway, I guess I'll try reading that Kan extension thing now that the demons keeping me company and awake fell asleep themselves. Have a nice day/night/whatever!

what are you reading right now /mg/?
>hartshorne
>hatcher
name a more iconic duo

>>12576343
you mean ironic duo?

>>12576343
Openstax Chemistry 1. After this, I think I'll finish Munkres, chemistry is lame.

>>12574756
I did. But not really in the way y'all are talking about.

I don't understand why reduce that pic's context to only sexual. Sure, a pic of your legs in stockings is suggestive, but I see it more like a cosplay. I didn't get aroused, but I find it cute. And that's because it has your personal touch in some details.
Beauty isn't entirely inherent, it comes partially manufactured in presentation and display. I like how you disposed your legs, the book by the side, your calligraphy, etc. And your legs look fine. Of course context was a pre-existent factor, I enjoy your personality and enthusiasm in general.
I also see no problem regarding loyalty, it's just an anonymous pic in an anonymous board ffs.

Don't let the mean comments and your ruthless self-critical side overwhelm your views and sense of self.

And yeah maybe posting it wasn't a good idea anyway (even though some of us liked), I mean, in your place I wouldn't manage to cope with all these meanies here. Take care, sis.

>>12576387
i actually meant moronic duo

>>12573870
seek help, unironically, seek help if this confuses you.
all polynomials have finite degree. they are not infinite sequences of coefficients. they are finite sequences of coefficients where the sequences have arbitrary finite length.

>>12575308
You are a paranoid rightist. Stop being a moron and stop having opinions about things that are outside your area of expertise. Imagine if all the political science majors came to you and started telling you about how Cauchy's integral formula is wrong.

>>12572772
It should work as long as you want to have H_fu\in L^2

>>12576395
Lad you’re simping for a tranny. Reevaluate your life choices.

>>12576499
Please return to wherever you came from.

taking an elective in algorithms and these people are struggling with the definition of big O

>>12576562
why is this surprising to you?

>>12573727
ok ill just hang on then
No I dont do further maths

>"What is the probability that two randomly selected people were both born on July 4?"
1/365 * 1/365 = 0.00000751

>What is the probability that two randomly selected people were born on the same day?
apparently 1/365 or 0.00274 according to my book

why are these answers different?

>>12576589
I've always heard the memes but i never thought I'd experience it in real life.

>>12576614
pick a person, how often is their birthday July 4th? Pick a second person, how often is their birthday also July 4th?

Pick a person, when is their birthday? pick a second person, how often is the second persons birthday the same as the first?

>>12576626
oooooooh that makes sense thanks mate

>>12576536
Sounds like I'm worshipping her, and that would be shameful except that here I'm no one in particular.
Nothing romantic though, I wouldn't want to date her, only be friends with.
I don't view gender dysphoria so negatively, in a way it's like calling having a 15 inches dick a disease. More than anything, "disease" is a label used to justify artificial scorn and sadism.

>>12575308
>and I'm worried that there will be no economic opportunities for mathematicians in the USA
What makes you think that?

>>12576640
You sound like the type that would take tranny dick in the ass.

Is there a chart or book list for someone who wants to specialize in number theory? What should I learn before that? I only know calc 1.

>>12576395
Aww, thanks for a sweet post! I guess I was a bit anxious and let things escalate in my head. It's so easy to let negativity take over, especially at nocturnal hours. Nice to hear you liked it, though. How is it going over there in Canada? Did you have a nice Christmas? Do you have any interesting stuff during the spring semester?

>>12576656
try ireland and rosen see how you get on

Trannies, kys

>>12571666
it's the opposite. the tricks are easy to pickup and the concepts are hard.

>>12576614
Because there's 365 distinct different ways that 1/365^5 event can happen: one for each day of the year, not just July 4th

>>12576683
Equivalently, given a number from 1 to 365, what's the probability you pick that number if you randomly choose from 1, 2, ... , 365

I'm having troubles showing that if [math]n[/math] is divisible by the square of some prime then [math]\mathbb{Z}/n\mathbb{Z}[/math] has a nonzero nilpotent element. Any tips?

>>12576773
Have you tried thinking about the problem?

>>12576778
yeah, it's me
I know that if such element [math]a[/math] exists then for some [math]m\in\mathbb{N}[/math], [math]a^{m}=0[/math], which would mean that [math]a^{m}[/math] is divisible by [math]n[/math], ie [math]a^{m}=nk[/math] for some [math]k\in\mathbb{Z}[/math]. With this I think I should pick an [math]a[/math] such that [math]a \vert n[/math], but I'm not so sure.

>>12576773
Decompose n into primes and consider their powers. At least one of them has power greater than one. One of those prime factors is important. Any more hints would probably ruin it completely.

>>12576656
I'll sketch something for you.

Pre-reqs 1:
One variable real analysis and an introductory course in algebra, linear algebra doesn't hurt either.

Introductory books:
Ireland's already cited; Ivan Niven's, Cox's "Primes of the form x^2+ny^2" is a great intro for the more algebraic (and modular) side of NT. There are others. Pic one, you don't want to spend too much time of your life on this stage. You can start simutanously with pre-req 1 courses.

Pre-req 2 (here I assume you learned most of elementary number theory from the previous stage).
For analytic number theory:
-Measure and Lebesgue integration;
-Complex Analysis
-(Optional) Something on Fourier analysis, harmonic analysis, complex geometry etc.
For Algebraic Number Theory:
-Groups, rings, modules, Galois theory... other undergrad stuff like topology
-Commutative Algebra helps a lot, though it's not so much necessary to start
-Notions of measure theory and integration, not that necessary but it helps in a few parts
For Arithmetic Geometry:
-All the undergrad algebra (groups, rings, etc), and a few others like topology
-Commutative Algebra
-A course in algebraic curves can help, see Fulton's for instace (not so necessary though)
For Modular Forms and Elliptic Curves:
-Complex Analysis
-Having a notion of basic algebraic curves helps
-undergrad courses of algebra, topology too

There's also additive number theory that I don't know much about, I think basic undergrad courses in combinatorics and probability would be enough. Knowledge on asymptotics would help a lot.

>>12576803
the max?

>>12576814
Maybe. I have no idea what you are trying to ask. You know how to write 0 as a product of primes. Look at the powers and ask yourself what you can do to one of the exponents because it has some extra in it. After you figure that out, try raising whatever you have to a suitable power and see what happens.

>>12576773
Have you thought about when this ring is a field? It's pretty much the same reasoning. Guess you didn't.

>>12576833
>when is a prime divisible by the square of a prime
guess you didn't either

>>12576773
to prove an 'if...then' statement, you should look at what you're given as an assumption, and try to use that to prove your conclusion. you can also try to apply the contrapositive or contradiction if the problem looks like it would favor those methods.

>>12576664
I have these sort of problems too, but usually it starts right at the beggining of the day, it doesn't get worse by night.
I'm glad to see you welcoming positive words.
>How is it going over there in Canada? Did you have a nice Christmas? Do you have any interesting stuff during the spring semester?
Hm am I supposed to be that number theorist / geometer you spoke about a few days ago? How curious haha
Christmas was ok, nothing special. I got chocolate and a wallet (was in need for one actually).
I'll probably be taking a course in algebraic geometry next semester.

Let me ask you, how did you got more interested in algebra? Did it gradually happen before you could even think of taking a direction?

>>12575091
usually the positive p>0 is chosen as the representative of the equivalence class of unitarily similar primes

>>12576862
You didn't get it.

>>12576939
If p is prime then Z_p is a field, but that doesn't means that Z_n has a nonzero nilpotent elements for n, see Z_6.

>>12576968
The problem I mentioned is a if and only if, the relevant part is the other direction. How would you prove that it isn't a field when n is not a prime?

Is >>12575609 a decent proof? Thanks

>>12576990
don't enumerate your argument like that, it makes it hard to read.
other than that, it's a pretty standard proof.

>>12576904
No I was thinking about a Canadian poster who I for some reason thought you were. My bad if you are not from Canada. Nothing special means it's not superbad either, so that is good I think! Useful gifts are the best ones. Is the AG course going to be more on the classical side, or will it be a barrage of sheaves, schemes and whatever the geometers have in their toolkit?
>Let me ask you, how did you got more interested in algebra? Did it gradually happen before you could even think of taking a direction?
It just somehow clicked. At first it was total Hebrew to me, and I even failed the first course. Then I had to retake the exam, spent a lot of time trying to get a grip on the stuff, and that's when it started to make sense. Combining that with topology which I also liked, it was pretty obvious that I'd start going to the algebraic topology direction. However, I've never had too much spatial reasoning capabilities, so I am very weak at imagining the spaces, and that sort of makes me an algebraist who uses topological methods more than an actual algebraic topologist. Other options to be interested in would have been logic (which I sort of liked but not more than "sort of"), analysis (which I appreciate but had problems with figuring the tricks out) or some applied thing (eww!), so yeah. Are you trying to find your thing?

>>12576968
If n=6, then n is not divisible by the square of any prime. Your counter-example isn't actually a counter-example.

>>12577004
Oh sorry. Just learning proof writing in general. Thank you.

>>12577073
A tip I have for you is to include more English (or whatever language you are writing in). In essence a proof is a piece of literature so it should read naturally.

>>12577094
Thank you for the advice. Do you have any suggestions for practice?

>>12577102
Do proofs and then read them back to yourself. If it doesn't flow properly or needs more connectors then rewrite it. Honestly it doesn't matter what you're proving but this will probably help you keep proofs straight when you're doing harder stuff.

>>12577115
Thank you! Have a nice day anon.

>>12570637
bumping once more
I still can't figure it out...

>>12577139
what even is [k\alpha]

>>12577174
floor(k*a), the smallest integer <= k*a

>>12577027
>Is the AG course going to be...
I think it's going to be closer to the second, but I'm not sure how far it is going to get.
>Are you trying to find your thing?
I already made my first official steps towards a choice, but my paranoid and catastrophiliac mind won't stop pondering.
>It just somehow clicked
I envy you, haha. Well, I can say the same in terms of taste and interest, but I'm still really not sure if I'm more naturally capable for whatever area. My better grades all align to my main interests, but I feel it's mainly because I took it seriously and studied harder for those. I don't know.
Either way I'll just try to pay attention to potential inclinations and ask myself if my taste can align with what I do better.

>>12577218
>I think it's going to be closer to the second, but I'm not sure how far it is going to get.
Sounds scary. Enjoy your germs and sheafifications of presheaves! I hope you know your tensor products well. I think the induction functor is one of the great 6, but I don't know. I steer away from scary stuff.
>I already made my first official steps towards a choice, but my paranoid and catastrophiliac mind won't stop pondering.
And that would be algebraic geometry or topology? At least you give hints that point that way.
>I envy you, haha. Well, I can say the same in terms of taste and interest, but I'm still really not sure if I'm more naturally capable for whatever area. My better grades all align to my main interests, but I feel it's mainly because I took it seriously and studied harder for those. I don't know.
I think that being interested in something is extremely important to be actually good at something. If you are bad at a thing you like, you have motivation to make yourself good. If you are good at what you like, you have motivation to improve. I'd say go for it, but that's just my perspective on things. But I think you should definitely know your tensor products just in case you change rings a lot in AG.

Is there a proof that even a single uncomputable number actually exists? When I see stuff like Chaitin's constant, it seems like the constant is actually a computable irrational, it's just that finding out which irrational is uncomputable.

>>12577332
>Sounds scary
I bet you can hold your ground on tensor products haha, but maybe you fear more the geometric stuff? (nah you're just bluffing)
I'm really just diving in, and I didn't even properly study classic AG yet. I hear it requires way more commutative algebra than classical AG, and I feel motivated by that. Already did a course in it, did well but still felt I didn't learn enough, so I've been reviewing it. I reviewed the basics of tensor products of modules and algebras again, with (very) basic categorical stuff along. Still feel it's not even close to what I'll need. What is the induction functor? Did a quick search and got results about induced representations of groups first.
>And that would be algebraic geometry or topology?
Algebraic Geometry. I don't have much affinity with geometry (yet!1), not sure how much of a geometric intuition I can develop, but for the longest time I had (have?) a mental block due to past experiences with crazy differential geometry courses. To be honest I wasn't much a fan of geometry (at least comparatively) even before, but I'm trying to get past traumas and experience it anew. The fact that it's algebraic and not differential is motivating (not that I dislike analysis, but many differential geometers sound like unintelligible physicists to me).
Thanks for warning me about tensors!

>>12577507
There are uncountable reals and countable computable numbers. In that sense, "almost all" real numbers are uncomputable.

i know this is slightly off topic but PLEASE SOMEONE
>>957023
HELP ME

>>12577537
/wsr/ post number 957023 PLEASE HELP I HAVE NO TIME
10 MINUTES NOW

>>12572556
Because an alternating multilinear map satisfying the Jacobi identity isn't the same as a positive-definite Hermitian form.

>>12577526
That's exactly the case! No bluff at all.
Yeah, I think there's a correspondence between the geometric notions and some algebraic notions (the details I do not know, though). Actually, I think it was an exercise in Matsumura's book to show that the (prime) spectrum of a ring is disconnected if the ring has an idempotent [math]0 \neq e \neq 1[/math], so similar connections with the geometric and annular world is to be expected.
>What is the induction functor?
Let's have everything be commutative and unital, so there are no stupid things to worry about now. If [math]S \le R[/math] are rings, then the inclusion makes the bigger ring a module over the smaller one, and so we obtain a functor [math]i_! \colon S\textbf{-mod}\to R\textbf{-mod}[/math] (or [math]i_*[/math] depending on the author) that is defined by [math]i_!(-) = R\otimes_S -[/math]. This is the "extension of scalars" functor. The inclusion isn't special in any way, though. Any ring homomorphism [math]S \to R[/math] would have worked, injective, surjective or neither. If it's surjective, then it feels wrong to say it extended the scalars, so I call it the induction functor. If you consider the group algebras where you include the algebra of a subgroup into that of the whole group, then you get what Google gave you. This is also the left adjoint mentioned in >>12573706 but the rings the other way around now.
>Algebraic Geometry...
I really do agree with you on that DG thing. It seemed so horribly messy with all the notation and all the DEs and that stuff. Having to play with atlases made it even more complicated for me. Luckily, such things are somewhat avoided by replacing differential with algebraic, BUT then you start getting all sorts sheaves and ringed spaces and what have you that should somehow carry local information, and I am not sure if it will be much easier after that. At least there is the whole field of commutative algebra to help you.

>>12577670
Oh I did see a few things involving this functor, but with a [math]S[/math]-module or [math]S[/math]-algebra in place of [math]R[/math]. There are some adjoint relations with sorts of forgetful, extension of scalars and Hom functors but apparently I already forgot details again.
How did you learn your category theory in order to apply in algebra? I didn't like how rushed of a introduction some algebra books gave of it, so I read a bit of "Category Theory in Context".
>At least there is the whole field of commutative algebra to help you
I want to try thinking in both ways whenever I can so I'll be able to complement when I get stuck in one side, but actually doing is a whole another story... I hope I'll do my best!

>>12577813
>[math]S[/math]-module
Actually no, I don't think that would make sense, this one lands on the same category. But maybe with the algebra? I'd have to check.

>>12577813
It is a functor between module categories. The ring homomorphism that induces it makes the codomain ring an algebra over the domain ring (not domain like an integral domain). The Hom would be a right adjoint to the restriction functor which is the one I was talking about in the old post, so you get a triple extension/induction - restriction - coextension/coinduction, which is two adjoint pairs. I think they should all be used in AG quite a lot.

I learned my CT from Awodey and Rotman's Introduction to Homological Algebra, and then just here and there while working on stuff. You will need homological algebra with all the derived stuff, so why not take a look? It will go through adjunctions and limits and that kind of things in a mostly algebraic setting, but you may also have examples of what things would be like for sets and other categories than just modules. His book also describes various types of rings fro a homological viewpoint. For example, let's say [math]R[/math] is von Neumann regular. Two equivalent ways to define it would be:
(1) For every [math]r\in R[/math], there is some [math]x\in R[/math] such that [math]rxr = r[/math].
(2) Every module over the ring is flat.
The first condition is nice if you want to check whether a field would be vNR, but the second one tells you how its modules do homological things. I recommend.
>I want to try thinking in both ways whenever I can so I'll be able to complement when I get stuck in one side, but actually doing is a whole another story... I hope I'll do my best!
That's a nice thing when you have a dual formalism. You will do your best! Victory or Valhalla!

>>12577846
Oh, and the forgetful functor could be from algebras to modules over a fixed ring.

>>12577813
>How did you learn your category theory in order to apply in algebra? I didn't like how rushed of a introduction some algebra books gave of it, so I read a bit of "Category Theory in Context".
My supervisor gave me Gelfand-Manin to learn CT. It was painful.

>>12576665
>>12576809
thank you very much

>>12577931
I said some bs, it was "restriction" when I mentioned extension. I've actually seen that this functor is left adjoint to restriction of scalars, which in turn has as right adjoint that Hom. But it was quickly mentioned in the statement of cancellation law and left adjoint for the tensor product functor by a fixed module, and I didn't worked out many applications yet (or I forgot).
Damn you sound like a wizard with all this CT, I hope I'll get to this level one day and then laugh at how impressed I got.
>so why not take a look?
Ok! Sounds resourceful for homology, too.
>von Neumann regular
I'd call "the lolifier ring". Sorry.
I swear I saw something similar about modules, like "every submodule is flat", but maybe I'm hallucinating.
>The first condition is nice if you want to check whether a field would be vNR
Every division ring is, and every domain that isn't a field isn't. Looks like a sudden jump. Is there any well known class of rings 'closer to the middle'?
(Also feel free to end our conversation for now, I'm already scared and embarrassed with things I'll have to graps and things I should know by now.)

>>12577972
Another homological algebra book? Geez

>>12578776
For your defence, you mixed up two left adjoints. But yeah, it's good that you are familiar with that triple. They will allow you to jump back and forth between the module categories, so I recommend you make yourself familiar with them!
>Damn you sound like a wizard with all this CT, I hope I'll get to this level one day and then laugh at how impressed I got.
I wouldn't say I'm super good with CT. I'm just not afraid to invoke categorical spells when possible, and thus I have some familiarity with this stuff. Honestly, sites like nlab make me very scared.
>Ok! Sounds resourceful for homology, too.
That's the main purpose of the book. That, Weibel or which ever you choose to follow, you will want to have a good grasp of derived functors, but those should be found in every intro book.
>I'd call "the lolifier ring". Sorry.
I see what you did there! Nice one.
>I swear I saw something similar about modules, like "every submodule is flat", but maybe I'm hallucinating.
That would be equivalent to "every module is flat".
>Every division ring is, and every domain that isn't a field isn't
Correct.
>Is there any well known class of rings 'closer to the middle'?
Semisimple rings. If we stay on the commutative side, a semisimple ring would be a product of fields. As every module over such a product is flat, that is also a lolifier.
>(Also feel free to end our conversation for now, I'm already scared and embarrassed with things I'll have to graps and things I should know by now.)
I will probably do so soon because I haven't slept for about 40 hours, oops. But at least you got some ideas what to check out, I hope.

Doing a course on some advanced topics in partial differential equations without having taken an "intro" to partial differential equations course due to some scheduling nonsense that means if I want to graduate on time I need to do well in this course. Done linearization and systems of ODEs before as well some "Turing morphogenesis" problems, so I might be okay but I have a particular question that may be very dumb:

What exactly is a rescaling? We do it a lot and I understand on some level that it's just convenient algebraic manipulation but it's got something to do with the method of dominant balance, getting your nonlinear term to scale in the same way as your linear term, but I don't quite understand exactly why it works out or why it's even justified to say "just let this be this times [math]\varepsilon ^2[/math] so that our linear term will be [math]\varepsilon[/math] so you can see the algebra works out"

Anyone know what I'm talking about or should I bust out some specific examples? I'm just looking for some kind of reading material or way of thinking about it beyond "just shut up and do what the professor says." Might not even be an issue insofar as on the homework he seems to have given us hints as to what rescalings to choose, but I wish I understood on some kind of higher level.

Hello bros I'm trying to prove this:

Let [math]x_t[/math] be a monodimensional Ito process
[math]x_{t+\Delta_t}-x_t=\int_{t}^{t+\Delta t}\mu(x_s,s)ds+\int_{t}^{t+\Delta t}\sigma(x_t,t)dW_s[/math]
with solution that exists and is unique. I want to show that
[math]\lim_{\Delta t \to 0}\frac{1}{\Delta t}\mathbb{E}[(x_{t+\Delta_t}-x_t)^2|x_t=x]=\sigma(x,t)^2 [/math]
So we have that
[math](x_{t+\Delta_t}-x_t)^2=\left(\int_{t}^{t+\Delta t}\mu(x_s,s)ds\right)^2+\left(\int_{t}^{t+\Delta t}\sigma(x_t,t)dW_s\right)^2+2\left(\int_{t}^{t+\Delta t}\mu(x_s,s)ds\right)\left(\int_{t}^{t+\Delta t}\sigma(x_t,t)dW_s\right)[/math]
the middle term should be such that by Ito isometry
[math]\lim_{\Delta t \to 0}\frac{1}{\Delta t}\mathbb{E}\left(\int_{t}^{t+\Delta t}\sigma(x_t,t)dW_s\right)^2=\lim_{\Delta t \to 0}\frac{1}{\Delta t}\mathbb{E}\left(\int_{t}^{t+\Delta t}\sigma(x_t,t)^2ds\right)=\lim_{\Delta t \to 0}\mathbb{E}[\sigma(x_{t+\Delta t},t+\Delta t)^2|X_t=x]=\sigma(x,t)^2[/math]
But I have no idea how to show that the other two are zero (if they are). Any idea?

>>12578998
I'm a bit sceptical that the (mu)^2 term would be zero, but I'm purely going by intuition here.
How about you wrap the E[] around them as seen where it goes?

>>12576499
>>12576640
dilate
>>12575970
based

>>12579050
I'm quite skeptical about it too, but I am quite sure that the result I should prove is true, but this one crams in a lot of things and I'm not having control over it right now

>>12579050
[math]\lim_{\Delta t \to 0}\frac{1}{\Delta t}\mathop{\mathbb{E}}[\sigma^2(W_{t+\Delta t}-W_t)^2|x_t=x]=\sigma^2[/math]
This is just a special case but it works

>>12574412
That's not (necessarily) how things work: the Planck length isn't some fundamental lattice spacing and space is not discrete. However we can use that as an approximation to do calculations, and in fact that is usually how the path integrals are defined/performed (start with lattice, then let spacing go to zero). Not sure about really advanced stuff tho, perhaps the other poster has more to say

>>12578860
>Honestly, sites like nlab make me very scared
The few times I went there for enough time I'd sort of think "Good, reminds me that even autism should have its limits." But maybe I don't know the true power of autism yet.

Because I'm seriously mentally damaged, I often say [math]x[/math] when (really deep) in mind I'm actually meaning [math]y\ne x[/math], which makes me sound extremely dumb. I did a bunch of times here, the possible last three are the restriction vs extension I mentioned, and the following two:
>That's the main purpose of the book
I meant to say "I could also use for homology when I need." (just a reaffirmation of its usefulness for me, this time from my end)
>That would be equivalent to "every module is flat".
I meant to say "A module [math]M[/math] is called [math]\cal{L}[/math] if, and only if, all of its submodules are flat." Let's call [math]\cal{L} := \text{ lolicondom}[/math] (you know, all his subs...). The zero module is a lolicondom, also every lolifier is a lolicondom. Not every module is a lolicondom. I don't think every flat module a lolicondom, but I don't have an example right now.
Oh, I actually did see something like lolifiers, way closer than I thought. It's an exercise defining them by your second definition but in the commutative context (with unity, all rings), though the name given is "absolutely flat". Proves that a ring is one iff its finitely generated ideals are direct summands, iff its finitely generated ideals are idempotents, iff every principal ideal is idempotent. There was at least some truth in the voices in my head!
(I know you didn't mean anything, it's partly a form of venting to feel less shitty about myself)
>Semisimple rings
Cool. One step down closer.
>I haven't slept for about 40 hours, oops
You tried and couldn't? Oh my, take care.
I should be sleeping, but I can't say I tried...
>But at least you got some ideas what to check out, I hope.
Yes! Thanks.

>>12578860
>Honestly, sites like nlab make me very scared
The few times I went there for enough time I'd sort of think "Good, reminds me that even autism should have its limits." But maybe I don't know the true power of autism yet.

Because I'm seriously mentally damaged, I often say [math]x[/math] when (really deep) in mind I'm actually meaning [math]y\ne x[/math], which makes me sound extremely dumb. I did a bunch of times here, the possible last three are the restriction vs extension I mentioned, and the following two:
>That's the main purpose of the book
I meant to say "I could also use for homology when I need." (just a reaffirmation of its usefulness for me, this time from my end)
>That would be equivalent to "every module is flat".
I meant to say "A module [math]M[/math] is called [math]\cal{L}[/math] if, and only if, all of its submodules are flat." Let's call [math]{\cal{L} }:= \text{ lolicondom}[/math] (you know, all its subs...). The zero module is a lolicondom, also every lolifier is a lolicondom. Not every module is a lolicondom. I don't think every flat module a lolicondom, but I don't have an example right now.
Oh, I actually did see something like lolifiers, way closer than I thought. It's an exercise defining them by your second definition but in the commutative context (with unity, all rings), though the name given is "absolutely flat". Proves that a ring is one iff its finitely generated ideals are direct summands, iff its finitely generated ideals are idempotents, iff every principal ideal is idempotent. There was at least some truth in the voices in my head!
(I know you didn't mean anything, it's partly a form of venting to feel less shitty about myself)
>Semisimple rings
Cool. One step down closer.
>I haven't slept for about 40 hours, oops
You tried and couldn't? Oh my, take care.
I should be sleeping, but I can't say I tried...
>But at least you got some ideas what to check out, I hope.
Yes! Thanks.

>>12576646
The fact that there are barely any at the moment, as far as I can tell. The people I know who are further along than I am are literally wondering when people will die so they can get jobs.

I think universities, because they are a vertex in the education-news-government triangle, could become a really nasty place to be during the period when I'm going to try for my Ph.D. I envision 1968 type activity.

More importantly I have no fucking clue how I would get post docs in Singapore, HK or the other places I'm considering. I know I'm being retarded but I'm really freaking out; a year of paramedical terror being inflicted gleefully on everyone I know by both officials and peers has left me a nervous wreck

>>12575632


fuck


you


faggot

what are the best chapters of Éléments de mathématique? I quite like the works of Serre and Lang but have only read things published under their own name.s

>>12579788
>The few times I went there for enough time I'd sort of think "Good, reminds me that even autism should have its limits." But maybe I don't know the true power of autism yet.
I'm pretty sure the nlab level requires bathing in blood at moonlight, or something similar. Black magick and following the example of Elizabeth Bathory could help reach that level.
>Because I'm seriously mentally damaged
I wish you could have your head checked. It really seems to cause you anxiety as you mention it quite often.

You are correct about not every flat module being a lolicondom. However, if you would have, I think, a Dedekind domain as your coefficients, that should suffice to make submodules flat as well. The exercise you are talking about justifies the alternative name for the lolifiers: von Neumann regular = absolutely flat. Sometimes the voices are friends.

>You tried and couldn't? Oh my, take care.
Yeah I tried but stupid thoughts kept me awake, and then I stayed up all day to avoid ruining my sleep cycle completely. Sleep well over there!

>>12580056
>1968 type activity
what happened in 1968 besides the 1 year anniversary of the USS Liberty attack?

>>12580197
1968 was marked by student "demonstrations" in the USA. Despite the innocuous name these were accompanied by extensive rioting, attacks on students and profs, and terroristic bombings, which historians justify by noting that they "did not target people, only property". Of course, the year is not so evocative since these types of events occurred sporadically throughout the 60s and 70s

>>12570637
ok one last time

>derivatives are just a bunch of rules
>theyre taught in Calculus
Why arent derivatives taught earlier? Theyre not hard

>>12580752
they are in other countries

>>12580056
Aren't you only talking about uni positions? If so, I think this isn't news and maybe it's not even getting much worse by now. But regarding the private sector, I'm skeptical. The US is one of the best places to find companies looking to hire math phds to solve their complicated business problems. Why would it be getting worse now?

>>12580544
You're either trolling or reaching unimaginable levels of brainletism, stop the spam.

>>12577528
>countable computable numbers
This is not true in any meaningful sense.

>>12580752
Do normogroids only ever apply rules memorized from some shitty textbook to do math? They really never stop to think "damn, there could be some deeper or simpler intuition about this"?

>at a conference in California
>woman
>need to use restroom
>woman coming out says there's a man in there
>go to see for myself
>first stall is open
>there's a guy inside peeing standing up
>he turns and growls at me, like I did something wrong
>I recognize this guy
>it's fucking Shing-Tung Yau
>stare in shock as he blows past me
>didn't wash his hands

>that night at conference dinner
>sit with male colleague
>says he saw Yau in the bathroom and he didn't wash his hands
>reply that I know

>>12581287
shut up tranny

>>12581287
He was angry you didn't use the right stalls, troon

>>12581140
I'm a stupid normie and I do but I'm stupid so what do you want me to do about it

>>12581406
A stupid person doesn't know he's stupid. If you just solve abstract math problems for a while, you'll get into a habit of finding different intuitions to understand a problem, because that's most often the only way.

>>12580165
>I wish you could have your head checked
I remember what you said. I'm considering.
>However, if you would have, I think, a Dedekind domain as your coefficients...
Curious. What you think of Dedekind domains? Do you often toy with them?
>I stayed up all day to avoid ruining my sleep cycle completely
Based & consistencypilled.

Possibly a brainlet question, but how was it figured out the the definition of continuity (pre-image of open sets is open) is a good one? Did they just realize that it was true for metric spaces, so decided to make it a general rule, or is there more going on?

How do you guys revise math? I read that simply re-reading is not a good way to remember stuff, and that active recall is much better. The problem, is I don't know how to set up a good revision system.

I have no problem remembering computations, but I tend to forget (the less important) theorems, only the main things stick with me. Even the main theorems of a subject can become lost after 1-2 years of not using them.

I would like to set up a rather robust revision system. Flash cards don't work well for math, I don't think, at least.

How do you guys revise?

>>12581596
for spaces with no metric it can be difficult to see when objects are "near"
while any two elements share at least one open set (the whole space), they can be thought of as being "near" if they are both elements of "many" open sets

>>12582011
By doing research and solving textbook problems. Use it or lose it.

>>12582011
>after 1-2 years of not using them.
there's your problem

>>12582015
Ahhh, right. That makes sense, thanks.
>>12582011
I don't really; I'm too slow, so if I were to spend my time revising, I'd do nothing else. I usually remember most things one way or another, and if I do forget some prerequisite, I just reacquaint myself pretty quickly.

>>12570637
Ok, now I think I finally fucking figured it out.
It holds for any irrational a, not just for 0<a<1.

This inequality is true:
[math] |a - \frac{p}{q}| < \frac{1}{qq'} [/math] where q' is the denominator of the next convergent ([math]q < q'[/math] holds).
and we're also given that [math]k \leq q[/math]

So we get [math] |ka - \frac{kp}{q}| < \frac{k}{qq'} < \frac{1}{q'} [/math] (*)

Case 1: [math]\frac{p}{q} < a[/math]
(*) becomes [math]\frac{kp}{q} < ka < \frac{kp}{q} + \frac{1}{q'}[/math]
Thus, it suffices to show that [math]\frac{kp}{q} + \frac{1}{q'} < [\frac{kp}{q}] + 1[/math]
Equivalently, show [math]\{\frac{kp}{q}\} < 1 - \frac{1}{q'} [/math] where {x} is the fractional part of x. (Notice that [math]\{\frac{kp}{q}\} = \frac{(kp) \% q}{q} [/math] where % is the modulo operation)
Equivalently, show that [math](kp) \% q < q - q/q'[/math] which is true since [math](kp) \% q < q - 1< q - q/q'[/math]

Case 2: [math]\frac{p}{q} > a[/math]
(*) becomes [math]\frac{kp}{q} - \frac{1}{q'} < na < \frac{kp}{q}[/math]
Thus, it suffices to show that [math][\frac{kp}{q}] \leq \frac{kp}{q} - \frac{1}{q'}[/math]
Equivalently, show that [math] \{\frac{kp}{q}\} \geq \frac{1}{q'} [/math], which is true since [math]\{\frac{kp}{q}\} \geq \frac{1}{q} > \frac{1}{q'}[/math]

Is it correct??

>>12580890
In the private sector I think there will be an economic contraction similar to '07 soon, due to COVID lockdowns' lasting impact (and other factors).

>>12581540
I don't remember doing anything with Dedekind domains in particular outside a quick course on algebraic number theory I had 4 or 5 years ago (how is everything so far away in the past?), but I think that the defining property that non-zero prime ideals are maximal is quite nice an idea. This makes its Krull dimension easy to tackle. I'm currently trying to see how I could fit a veeeeeeeeeeeeeeeeeeeeeeeeery general thing my supervisor came up with his buddies in a pub (literally) to ring theory. I know everything works just fine for those lolifiers, so maybe I could try Dedekinds next, as that would be from Krull dimension 0 to 1. However, there's the nasty remark you made that an integral domain (which these are) is von Neumann regular iff it is a field, so I can't have both properties simultaneously if I want something interesting to maybe take place. Thanks for the idea!

>>12582011
To be honest, I remember pretty much only what I work on and with, and to revise stuff I start using it and learn while trying, failing and succeeding.

>>12582192
good for you lad!

Recommend me a good book to learn algebra from. I've really only heard of Dummit and Foote, but I'll be using it for self-study, so answers to exercises is key. Also, I've heard that D&F is more encyclopedic in scope, which I don't feel is particularly good for a first brush with the subject.

>>12582796
Pinter

>>12582796
There are no answers to the exercises therein, but Clark's Elements is almost perfect for self-study. Luckily if you're doing algebra at a learner's level, you'll very rarely make errors in proof attempt which aren't obvious at first inspection.

>tfw no tranny gf to discuss ring theory with

>>12583262
No problem with you wanting to hug a tranny but please at least recognize that "trans woman" is a type of man

Daily reminder: algebra is for trannies and faggots

>>12583818
And analysis is for basedkikes and jews and topology is for curry eating indians and niggers. What can we do?

>>12581287
There are some cultures where it's customary to wash before and not after. That's what I do

>>12581287
>>12583925
Please "bee" understanding, LOL.

>>12583922
Geometry.

>>12583989
Algebra is but written geometry and geometry is but figured algebra.
t. Sophie Germain

>>12583818
and the rest of math is for autists and sperglords

>>12583989
Geometry is for Eminem-bumping wiggers and toothless white trash, anon.

>>12584155
As well as shifty gook types.

>>12584039
Why would you trust a woman's word on geometry?

>>12584333
Honestly, I would trust the word of anyone except those who say geometry is intuitive.

>>12584374
Well, you are a tranny so your brain is wired like a woman's. To have a geometry-sense is something ineffable for you. It's natural to be skeptical.

2d - intuitive
3d - fucking mess that will fuck with your brain
anything beyond 3 dimensions - made up garbage which doesn't exist it's all a ruse of jewish topologists

>>12584418
It's all intuitive to the white man, 2d, 3d and above. See: Coxeter.

>>12584390
I guess. It would be nice to be able to see all sorts of twists and stuff in spaces with my inner eye, but nope. I have no idea what a degree n map would do, like really do, to the 2-sphere, for example.

>>12580092
Almost everything is unreadable due to unending cross-references
Just read Lie algebras 4-6 (self-contained stuff about Coxeter groups and hyperplane arrangements), possibly Algebra 8 (the revised version) and maybe Algebraic Topology and Spectral Theory.
Basically only read their self-contained volumes

>>12584418
Literally all the intuition you need. Ask any topologist to clarify what they mean with an example, and 90% of the time they'll draw a torus.

>>12584491
You could just write one down? Animate it if you have to.

>>12570292
yes

>>12585380
I tried googling ages ago but found no formula at all. Even if I did, I would have no idea how to animate it.

>>12585460
Aw hell I'll do it myself and post it.

>>12585583
Okay pic related is a degree 2 map for the 2-sphere. To get a degree n map on the 2-sphere it looks like you just take a degree n map on the circle and suspend it.

>>12585636
Forgot pic.

>>12585583
>>12585636
>>12585641
Thanks! I thought it would probably be something like that but then I kept worrying about ripping the sphere. How did you do that, by the way?

>>12585675
That came from the Wikipedia page on degree of a map, but you can use a parametric plot in Sage if you just have a formula for the map you want. You just have to let the radius vary so you can see the winding.

>>12585701
Maybe I should try to learn how that Sage works, then. Many thanks, my friend.

>>12585766
No problem, have fun!

https://iro2.net/mathematics/tljwzarw/

>>12573606
I can't tell which one is the arc. From the context I know it must be the green one but my brain tells me its the black one.

>>12575828
18 is approaching under weight

>>12582307
I pictured an outgoing not-so-old, but maybe your supervisor just hangs out with peers from time to time. When are you meeting him? It sounded like you sort of promised stuff until a date.
Did you enjoy that algebraic number theory course?

>>12582011
Up until now I just reviewed for tests by making a summary and trying to remember related stuff going point by point.
I think I'll have to improve this a bit for my qualifying exam, I'm thinking of trying to synthetize ideas by thinking on how to explain proofs or everything clearly in a few words and then maybe drawing some maps/diagrams that can serve as a skeleton of the basic ideas etc. I don't know how to do these things, it's going to be a new experience. And this is nothing new, you'll find people talking about mental maps or developing simpler ways to teach what you learn as a way to consolidate your learning (Feynman is an example of someone who talked about this last practice).

>>12586266
I actually checked it. There's a new formula that takes the height(/length? Is a human high or long?) into account and the calculator gave 17.6 classically and 16.7 in the new way. Winters are a bit chilly, to be desu.

>>12586277
>I pictured an outgoing not-so-old, but maybe your supervisor just hangs out with peers from time to time
60 this year. I hope I get to see him live around that, so that I may give him a present.
>When are you meeting him? It sounded like you sort of promised stuff until a date.
I think the semester starts next week or the one after that, so probably then. I promised to try get some examples done, but no luck with that so far.
>Did you enjoy that algebraic number theory course?
If I remember correctly, somewhat yes. Not the best thing ever, but it was nice to get some more motivation for ring theory (many parts of which are essentially created to solve number theoretical problems).

>>12586310
Bmi = weight(kg)/height(m^2)
My bmi = 86/3.46 = 25.1

>>12586318
3.42*

>>12586310
BMI has always taken height into account, or at least for a long time. I don't know if a trans woman should use the female or male table though.
Cute picture.

>>12586319
Anon are you 3,5 meters tall?

>>12586318
Only now saw the ^2.

No it's taking the square, which means I'm 1.85 meters tall.

>>12586318
Yes, that's the classical. Oops I made a typo when calculating the original one. Should have been 62/3.57 = 17.4.

>>12586324
It does, but the new ones is less horrible for tall people and hurts midgets more. I doubt the BMI itself has anything to do with being male or female. https://people.maths.ox.ac.uk/trefethen/bmi_calc.html

>>12586343
Then it must be only body fat percentage that has to do with sex.

>>12586352
This is why ffmi is better.
https://www.calculators.org/health/ffmi.php

>>12585675
just compose stereographic projection with polar coordinates, the mapping phi -> n*phi has degree n

>>12586343
So you're 1.88m tall? Sounds ridiculously tall for a finnish person.

>>12586343
So you're 1.88m tall? Sounds ridiculously short for a finnish person.

>>12586352
That and skeleton. The male skeleton is much more robust and thick. Consequently, females have more osteoporosis cases as the skeleton starts to break down with age.

>>12586366
That requires knowing your bodyfat first, though.

>>12586376
Oh yes of course! Around the origin that would do nothing and ridiculously far that would spin awfully much, and thus be compatible with the infinity point and then the poles. Gotcha! Thanks.

>>12586392
Yes... My dad is 10 cm shorter than me. It sort of stings a bit.

>>12586310
>I hope I get to see him live around that
Is he healthy? I hope you can give it to him.
>but no luck with that so far
Must be tough, but you seen to drive well the challenging research road. I'm not even there yet and I'm already unstable.
>but it was nice to get some more motivation for ring theory
It was very nice to see all the abstract algebra I learned arising naturally, being useful and effective to, ultimately, give nontrivial results about integers.

>>12586638
>Is he healthy? I hope you can give it to him.
He seemed quite strong and vivid before Christmas, so I hope so. As everything is done online, he doesn't actually live in my city at the moment, so I don't know how the virus situation is over there. Assuming the uni is reopened over the course of the upcoming spring, he will come back and I can give him something.
>Must be tough, but you seen to drive well the challenging research road. I'm not even there yet and I'm already unstable.
The worst thing, as you will find out later, is not hard material but the stupid voice saying that you haven't done enough and you could have done things better. I'm not sure if everyone has this, but my mom and I have it really loud and you seem very likely to have this nagging voice in your head as well, so it is worth it to warn you. It is not a completely trustworthy source of information, though. It told me for months how I'm useless and should quit, but then I got a lot of praise in my first year assessment, so it did make a verified mistake once. Please do not let your own version of that voice demoralise you. It will definitely try to
>It was very nice to see all the abstract algebra I learned arising naturally, being useful and effective to, ultimately, give nontrivial results about integers.
Exactly! There is a reason why the type of rings where there are no zero divisors are called INTEGRAL domains and not just domains, there's a reason why ideals are called ideals, and why some of them are prime. All from number theory. Some integers just happen to be ideal or prime!

This was a good episode of /mg/.

Not sure if this is the right place to ask but I'll go ahead...

So I'm in high school, and I want to do mathematics once I graduate. However, there's this one thing that bugs me... I'm super interested in advancing my mathematics knowledge, and I study it every day. I finished calc in my first years of HS and now I'm doing more advanced subjects. I also score really well in exams and tests at school (some of the best in the entire state/nation). However, I am really quite bad at competitive math, so-much-so even normie students can out-rank me. I am in math groups irl with some of the greatest minds, and they're all amazing in competitive math (IMO level) whereas there's little me getting shitty scores... I'm only seen as their superior in grades and exams, and also since I know more advanced math but I still feel sadness every time I think about how I can't even solve basic competitive problems... Do I have a future in maths (especially in pure maths, my passion (although applied isn't the worst))? I really want to do great things in the field but it seems like everyone in it that is really influential also competed in the IMO in high school, or something else.

Also side note: the Putnam exam is easier for me due to it being heavy on calculus etc. which I'm seen as a "king" of at school. It's just the number theory stuff I dislike and do poorly in. Any advice (esp. from math majors) would help me so much anons, thanks.

>>12587766
If you want to do pure math then I mean you're comparing problems made to be answered in a short amount of time compared to math that has to be researched for a long period of time.

>>12587766
nah competitive maths is a different kettle of fish
you will find the topics you enjoy in higher education

>>12587771
I did some light research (online for a university program) with the help of programming on Collatz sequences and generalisations with some proofs of trivial cases, so yes I love research. I also have a single paper written on a topic in classical mechanics, not sure physics is for me though.

>>12587772
Do you think I'll be able to have a career in maths without a career in competitions? I think the reason so many past-IMO competitors become great mathematicians is correlation with a passion for maths and not due to comp. maths giving them some kind of "problem solving superpower"?

>>12587782
probably not the next gauss
but you can try

New >>12588999