/mg/

[Anonymous No.12147606]

Arachnophobia Edition.
Previously on /mg/: >>12130804

[Anonymous No.12147612]

wrong thread link moron
should be >>12139430

[Anonymous No.12147628]

>>12147606
>Elliptic curves

[Anonymous No.12147637]

>>12147505
Apologies then brit(?)friend

[Anonymous No.12147642]

Nikolaj belongs in a grave.

[Anonymous No.12147643]

>>12147606
I don't get it, wouldn't that be Arachnophilia?

[Anonymous No.12147648]

>>12147606
>>12147643

[Anonymous No.12147655]

>>12147643
Listen, I was just trying to make a thread before someone else made one and we ended up with 2.

[Anonymous No.12147946]

>>12147606
>tfw he just became a politician
>got elected in french congress in 2017
>campaigned for mayor if Paris this year
>lost
>probably hasn't done any maths in 3 years

[Anonymous No.12147976]

>>12147946
mathematicians should stick to maths

[Anonymous No.12148003]

>>12147976
At least people in France can now identify 3 mathematicians:
Pythagoras, Descartes and Villani

[Anonymous No.12148061]

>>12148003
You forgot Lagrange. He even has a street named after him in Paris.

[Anonymous No.12148103]

Is metamathematics a thing that you actually study or is it just something that comes up when you study logic and axiomatic set theory

[Anonymous No.12148111]

>>12148061
Lay people still don't know who he was.
At least they can't cite him from the top of their head, maybe if they hear the name they might recall that he was "some sort of scientist, maybe a physicist".

[Anonymous No.12148251]

Might be an open problem:
Lets [math]s(n)[/math] denote the sum of digits of [math]n[/math]. Find all such [math]n[/math] that [math]s(5^n)=s(2^n)[/math]

By looking at remainders mod 9 we can come to the conclusion that [math]n=3+6\cdot k,\ k\geq 0[/math]. By putting [math]k=0[/math] we get one and, by my hypothesis, the only solution. Where can we go from here? It would be sufficient to prove that [math]s(8\cdot 64^k)<s(125\cdot 15625^k)[/math]. How do we go about that?

[Anonymous No.12148323]

>>12147642
I'm not save from the death drive - the End sounds like a comfy place to be in.

[Anonymous No.12148328]

>>12147606
Extremely based OP.

[Anonymous No.12148370]

Do you think Cedric the Villain is trying to establish an arachnomonarchist France?

[Anonymous No.12148377]

>>12148003
Everyone has heard of Poincaré

[Anonymous No.12148390]

Cute puzzle:
In a classroom each student decides to play a game where half of students always lie (naughty) while the other half always tell the truth (truthful).
The teacher knows hears this and asks John whether hes truthful or naughty. John answered but the teacher has difficulty hearing and didnt hear the answer, so he asked Mark and James to tell him which one John claimed to be. Mark answered "Naughty" while James answered "Truthful".
Question:which one of Mark and James is truthful and which is naughty?

[Anonymous No.12148412]

>>12148390
john can’t say naughty
so James is truthful and Mark naughty

[Anonymous No.12148415]

>>12148412
Yah but now you ruined it for others.

[Anonymous No.12148421]

>>12148415
hm you are right I will delete

[Anonymous No.12148427]

>>12148390
3 mathemagicians go to a restaurant for a lunch, and after they have finished eating a waitress asks if they all want coffee. Each of the mathemagicians answers "I don't know" and she makes some deductions of her own. How many coffees does she bring to the table?

[Anonymous No.12148431]

>>12148427
If they answered one after the other then one of the mathematicians is lying :)

[Anonymous No.12148437]

>>12148431
Blind or deaf *

[Anonymous No.12148442]

>>12148437
A deaf mathematician wouldn't have heard the question, and so wouldn't have answered anything

[Anonymous No.12148447]

>>12148103
Depends on what you count as mathematics.
For example, if you define math as "proving theorems about mathematical objects", and accept the Curry-Howard correspondence, then work on denotational semantics might qualify as a metamathematical thing to study.
>>12148427
Does the waitress needs to use excluded middle for her deduction?
I can't see any way to do it without LLPO at least.

[Anonymous No.12148449]

>>12145426
Still wondering.

[Anonymous No.12148451]

>>12148431
Very good.

>>12148447
She is not autistic, so she accepts LEM.

[Anonymous No.12148479]

>>12148451
>She is not autistic
But some of the mathematicians may be, and if even one of them (call him Anon) is constructivist enough for his coffee-wanting proposition to be undecidable in his internal logic, then the waitresses's conjunction is undecidable even classically, and so the mathematicians can consistently answer "I don't know" even if they speak in sequence.

The waitress who accepts LEM is effectively deciding for Anon that he wants coffee, which is a consistent solution but not necessarily the unique one (i.e. the situation is not model-theoretically categorical).

[Anonymous No.12148525]

>>12148449
Excluding the boundary spheres, the characteristic maps of the cells are homeomorphisms. Whatever this [math]\pi^{-1}(N)[/math] looks like, it deformation retracting to [math]X_{p-1}[/math] should guarantee similar activity for its preimage and the boundary sphere in each of the balls. Then do as the other anon said to obtain the horizontal isomorphism. I'm not completely sure, though.

[Anonymous No.12148736]

>>12148525
>should guarantee similar activity for its preimage and the boundary sphere in each of the balls
that's exactly the step of which I think that it doens't work

[Anonymous No.12148761]

Are mathematicians capable of telling me how to find the first integral of a physical system if the change in energy of the system depends on its state?

[Anonymous No.12148809]

hi
baby stats here. Not sure if this is the right place to post this.
Writing a python program to assist me my stats unit, currently trying to calculate PDF without using the N.D. tables, not sure if I'm doing it right.
1/(lcaseSigma * math.sqrt(2*math.pi)) * (math.e**(-1/2 * (zScore)**2))

Those who aren't familiar with python a ** b is the same as a^b.

[Anonymous No.12148869]

>>12148736
Have you tried it? A quick brainstorming gave me this idea: If we define [math]M := \Phi_a^{-1}( \pi^{-1}(N))[/math], then you could maybe define [math]H\colon M\times I\to M, H(x, t) = \Phi_a^{-1}(h(\Phi_a(x), t))[/math] for [math]x\not\in S^{p-1}, t<1[/math] and [math]H(x, t)=x[/math] otherwise. Could this be extended to [math]t=1[/math]? Perhaps. If so, then [math]H(x, 0) = \Phi_a^{-1}(h(\Phi_a(x), 0)) = (\Phi_a^{-1}\circ \Phi_a)(x) = x[/math] even for the interior points and this would then give you the desired homotopy. Here [math]h\colon N\times I\to N[/math] is the deformation he mentions in the text.

[Anonymous No.12148906]

>>12148869
>mfw zoomers say brainstorming instead of "thinking"

[Anonymous No.12148921]

>>12148761
so you have
[math]y'=f(y)[/math]
and need
[math]\int y dx[/math]
???

[Anonymous No.12148928]

>>12148906
Imagine being almost 30 and still a zoomer.

[Anonymous No.12148942]

>>12148869
This was also my first idea. The problem is that [math]H(x,t) [/math] is not well-defined for t>0 since [math]h(\Phi_\alpha(x),t) [/math] lies outside the domain of [math]\Phi^{-1}_\alpha [/math]

[Anonymous No.12148973]

>>12148921
The first integral of a physical system is not what you think it is, algebraist.

[Anonymous No.12148974]

>>12148973
>The first integral of a physical system is not what you think it is
this

[Anonymous No.12148988]

>>12148928
All the cons of being a zoomer while still being older than twenty four.

[Anonymous No.12148996]

>>12148390
No matter what John is, he said truthful, so mark is naughty and James is truthful

[Anonymous No.12149044]

>>12148942
Really? I'm not saying you are wrong, I just can't make myself see it at the moment. At least we know that [math]\pi^{-1}(N)[/math] can't have any extra holes because that would give non-isomorphic homotopy groups contradicting [math]X_{p-1} \simeq \pi^{-1}(N)[/math]. Then [math]M = \Phi_a^{-1}(\pi^{-1}(N))[/math] can have at most one hole. If it has none, then [math]M = D^p_a[/math]. This would then allow us to deformation retract the cell we are attaching to [math]X_{p-1}[/math] via [math]\Phi_a[/math] onto [math]X_{p-1}[/math], not nice. This will then force [math]M[/math] to be a fattened (and maybe a bit odd looking) version of the boundary sphere. In this case, it will follow that [math]M\simeq S^{p-1}[/math], as desired. I'm sorry if this is a bit shaky. I've been feeling like a vegetable all day.

>>12148988
Must you hurt me like that? Must you cause me pain unneeded?

[Anonymous No.12149052]

>>12149044
Oh and something to add: here I'm just assuming [math]X_{p-1}[/math] to be connected. If it is not, that's not a big deal, you do the similar analysis for its components and so on.

[Anonymous No.12149058]

>>12147606
whomever that is he is too british to be french

[Anonymous No.12149102]

>>12148906
it's "her" female brain

[Anonymous No.12149141]

>>12149044
>Then M can have at most one hole
?

[Anonymous No.12149173]

>>12149058
he's a real fancy boy

[Anonymous No.12149190]

>>12149058
dude looks ridiculous. If youre gonna wear a suit at least get a nice suit not a cheap piece of shit

[Anonymous No.12149196]

>>12149141
We don't care about what [math]X_{p-1}[/math] actually looks like. We know that [math]\Phi_a^{-1}(X_{p-1}) = S^{p-1}_a[/math]. On the other hand, if there would be two or more holes in [math]M[/math], these would contribute to [math]\pi^{-1}(N)[/math] as at least one extra hole which would then turn the corresponding homotopy group of [math]\pi^{-1}(N)[/math] into something that is not isomorphic to the same homotopy group of [math]X_{p-1}[/math]. This because the holes would be inside the ball, not on its boundary, and so the homeomorphism would preserve them because it would map them inside the attached cell. Is this complete non-sense? I'm starting to feel a crushing uncertainty.

[Anonymous No.12149219]

>>12149190
Imagine being filtered by Villani's shitty suit + shitty hair + shitty beard + spider aesthetic.

[Anonymous No.12149222]

>>12149190
i think it's on purpose anon. he looks like a fun type of person

[Anonymous No.12149226]

>>12149222
although that is some very british humor

[Anonymous No.12149237]

>>12149226
He's French anon, also I'm pretty sure he's of Italian ancestry as well.

[Anonymous No.12149279]

Retard here. Last winter I tried self studying Rudin's analysis and couldn't make it to chapter 3 and gave up. A few weeks ago I tried doing an easier book (Abbott's Understanding analysis) and maybe it's because the first chapter is stuff I already learnt, but it is too boring. I am thinking I should restart Rudin but use Abbott as a companion, so whenever I get stuck just go to Abbott (he actually has diagrams in his book). Is this a good idea? Or should I try something in between Abbott and Rudin?
Note: Whenever I got through a theorem on Rudin I honestly felt really good about myself and that I truly learned something new, and that's the feel I want again. Please help me. In case it matters, I did and undergrad and masters in EE, and am a phd dropout and just wanna learn math for fun (knew machine learning and DSP etc). The goal is to learn probability theory but mainly fun. Should I try something completely different instead?

[Anonymous No.12149329]

>>12148251
rent a supercomputer

[Anonymous No.12149337]

>>12149279
Try Tao.

[Anonymous No.12149390]

>If P is a point in 2D space with a Y coordinate of 2, then the point is (3,2)
Is the above statement considered false or "uncertain" in boolean logic?

>if P is a point in 2D with Y coord of 2, P is a kitty cat
Is the above statement false or considered meaningless? Points can't be kitty cats, but meaningless things could sensibly be considered false
---------------------------

I noticed that if a statement is always true, any other statement implies it. The way I think of it is that "implies" just means "in a reality state where P is true, Q is true" and given Q is true in all reality states, P just acts to "boot up" reality in the first place and shows that Q is true by mere existence.
--------------------------------

As far as the null/empty set, it has inspired me to come up with an interesting system pertaining to nothingness. It may also relate to a concept in set theory called "Forcing".
>A and B have no shared elements
>thus their intersection does not exist
>OR
>their intersection contains no elements
>if intersection does not exist, it can neither be in or not in A
>if the intersection does exist, being in A implies
>no elements are in A (false, A contains elements)
>no elements are not in A (which implies every element in the universe is in A, which is false)
>so the intersection cannot be in A or outside of A, regardless of whether it exists
>so it is "nin" A
The concept of nin is a new development. It explains the interface between nothingness and somethingness, without contradictory statements. As far as the statement that "nothing"/"no elements" are in A, that seems like a statement that nothingness can have a beingness trait (saying nothing IS inside is a conjugation of the verb to be). However, nothing can be taken in the form of a check on all existence returning no results. Nothing can "be" in that case, where it is free from interface with existence and "beingness" can be interpreted on the nothingness side of reality.

[Anonymous No.12149393]

>>12148427
>>12148431
And if they all answer at the same time then they all want coffee, so 3.

[Anonymous No.12149423]

>>12149196
yes. It's nonsense. You try to argue by some vague idea that Phi preserves holes which is useless because
>it doesn't take torsion into account
>you already assume that Phi induces isomorphisms on homology (if we ignore torsion)

[Anonymous No.12149432]

>>12147642
Nikolaj is my waifu.

[Anonymous No.12149439]

>>12149423
Okay. That's what I feared. Hopefully someone else can give you the correct idea then.

[Anonymous No.12149455]

>>12149058
Personally I like his style. I kind of yern for the days when people actually dresses much more formally in suits. It was much less degenerate than casual cancer that Sillicon valley created of jeans, t shirts, and flip flops.

[Anonymous No.12149466]

>>12148251
Theorem which might be usefull: [math]s(2^n)<3n+9[/math]

[Anonymous No.12149514]

Why is math always so focused around personalities?
The math itself should be in the focus.

[Anonymous No.12149522]

>>12149514
I study algebra, you think I would be able to understand even a tiny fraction of Cedric's work? Maybe if I studied mathematical physics for a few years.

Personalities exist because it's near impossible for different fields to have conversations.

[Anonymous No.12149529]

>>12149514
It's not really, it's more like those who choose to be eccentric stick out. I mean, certainly this makes them more interesting since your average mathematician dresses as NPC as possible.

[Anonymous No.12149532]

>>12147606
Martin Hairer or whatever his name is looks like a total chad except for his balding head.
The chin on that fucker is nuts.

[Anonymous No.12149546]

>>12149532
He has the gigachad chin lmao

[Anonymous No.12149553]

>>12148251
my thought was: [math]2^n \cdot 5^n = 10^n[/math] and maybe there's some trick around that, but i've found nothing
many of these "do shit with digits" problems are absolutely fucking hopeless
there is a natural idea "2^n has roughly n times [math]\log_{10} 2[/math] digits, and the digits are probably random enough" but no one knows how to do anything with this
https://oeis.org/A001370

>>12149514
let's be honest math is boring, so mathematicians search for other sources of entertainment

[Anonymous No.12149556]

>>12149553
>math is boring
kys yourself out of /mg/

[Anonymous No.12149563]

>>12149556
sorry but that's the objective truth...

[Anonymous No.12149570]

>>12149563
Math is better than sex.

[Anonymous No.12149574]

>>12148921
Negative, first integral is just a synonym for a conserved quantity.

[Anonymous No.12149597]

>>12149570
i doubt you've ever done any of those

[Anonymous No.12149605]

Can /mg/ clarify just what the heck is metric embedding theory all about, and what are its algorithmic applications?
(Assume familiarity with undergrad topology and approximation algorithms.)

[Anonymous No.12149679]

>>12149439
the absolute state of phd students

[Anonymous No.12149692]

>>12148427
3 because they don't know if the waitress wants coffee aswell and they are also autistic

[Anonymous No.12149697]

>>12149692
holy fucking shit

[Anonymous No.12149700]

>>12147606
dd

[Anonymous No.12149722]

>>12149679
Pls no bulli.

[Anonymous No.12149748]

>>12148061
Lagrange was italian you faggot.

[Anonymous No.12149785]

>>12149605
The rough idea is:
Let's say you have a metric space [math](X, d)[/math] and another metric space [math](Y, e)[/math].
Imagine that X is ugly and Y is nice for some reason - for example, Y is euclidean space.
If you had an embedding [math]f:X \rightarrow Y[/math] such that [math]d(x_1, x_2) = e(f(x_1), f(x_2))[/math] for all the [math]x_1, x_2 \in X[/math], then you could forget about X and just deal with stuff in euclidean metric.
Obviously that's wishing for too much in many cases, so people look for embeddings where [math]c_1 d(x_1, x_2) \leq e(f(x_1), f(x_2)) \leq c_2 d(x_1, x_2) [/math] for good constants c_1, c_2.

For example, imagine that someone comes up with a quick algorithm for travelling salesman problem in the special case when the cities are actually points on a plane, and the distances are just euclidean distances between the points.
Then, given a general instance of TSP, you could hope: maybe we can find points on a plane which have similar distances, and then run the quick algorithm, and that would give us an approximate solution - where the approximation constant depends on the quality of your embedding.

[Anonymous No.12149838]

https://youtu.be/UuRxRGR3VpM

I’m just gunna leave this here

[Anonymous No.12150064]

>>12149748
Cope.

[Anonymous No.12150103]

We have
[math]\Re\left(\frac{df}{dx}\right)=\Re\left(\frac{d\left(u+iv\right)}{dx}\right)=\Re\left(\frac{d\left(u\right)}{dx}+i\frac{d\left(v\right)}{dx}\right)=\frac{du}{dx}=\frac{d\left(\Re\left(f\right)\right)}{dx}[/math]
since both the real part function and the derivative are linear in their arguments, doesn't this mean that the matricies encoding [math]\Re[/math] and [math]\frac{d}{dx}[/math] commute? ie [math]\Re D=D \Re[/math]
Does that have any neat implications about the two functions?

[Anonymous No.12150133]

>>12149785
Is Chitosi /mg/'s official anime girl?

[Anonymous No.12150144]

>>12150103
>[/math]sinceboththerealpartfunctionandthederivativearelinearintheirarguments,doesntthismeanthatthematriciesencoding[math]

[Anonymous No.12150159]

>>12149279
Pugh is really good. It's got similar content to Rudin but is a lot more fun and visual and has better descriptions.

[Anonymous No.12150161]

Slow thread, so I'll post my favorite bait.
For [math]f, g[/math] Riemann-integrable in the interval [math][a, b][/math] and [math]f(x) > g(x)[/math] for all [math]x \in [a, b] [/math], prove that [math]\int _a ^b f(x) ~ dx > \int _a ^b g(x) ~ dx[/math]
>>12150133
No, Colette is.

[Anonymous No.12150168]

>>12150103
>Does that have any neat implications about-
no

[Anonymous No.12150221]

>>12149679
I don't think trannime is the most accurate representative of PhD students.

[Anonymous No.12150262]

>>12150221
You'd be surprised.

[Anonymous No.12150265]

>>12150168
:(

[Anonymous No.12150267]

Help /mg/ I'm choosing modules for my final year. My plans are to take a msc in cs afterwards, spend a few years in industry and then come back to a masters in maths and hopefully PhD.

I'm interested in information geometry/probability with differential geometry so the courses I've picked so far are

Measure theory
Manifolds
Topology
Probability theory
Computing in C

I get to choose from:

Functional analysis
Matrix analysis/Algorithms
Mathematics of machine learning
Math modelling with Pdes
Complex analysis
Theory of Pdes - more proof based pde theory
Numberical analysis with pdes

Of course there's algebra modules but I rather pick ones to do with what I want to do in the future

[Anonymous No.12150278]

>>12150262
I'm a grad student. We're basically all nerdy straight guys. I'll give you that there are some weebs and gays around, but I've never seen a gay weeb, much less a trap.

[Anonymous No.12150280]

>>12149279
if you want book recs: Buck's advanced calculus is good and for a pleb Introduction to Real Analysis by Bartle and Sherbert is probably more gentle

Tho desu you probably should just stick to Abbot
if you're just feeling bored with a section maybe check to see if you can do the exercises and move on if you can. it's not a novel, no need to read through every line.

[Anonymous No.12150296]

>>12150221
Sorry I can't keep my thoughts together at the moment for some reason.

[Anonymous No.12150319]

Suppose [math]Y[/math] is path-connected. I want to prove that every two continuous maps [math]f, g : [0,1] \rightarrow Y[/math] are homotopic.

Let [math]\gamma_{x,y}(t)[/math] be [math]x - y[/math] path in Y.
Define homotopy [math]H_t(s):[0,1] \times [0,1] \rightarrow Y = \gamma_{f(s), g(s)}(t)[/math]. Clearly, [math]H_0(s) = f(s)[/math] and [math]H_1(s) = g(s)[/math].
How do I show that [math]H_t(s)[/math] is continuous. I know that [math]\gamma_{x,y}(t)[/math] is continuous, but how do I show continuity by [math]s[/math] variable?

[Anonymous No.12150322]

>>12150296
It's ok, anon.

[Anonymous No.12150343]

>>12150319
That's obviously false.

[Anonymous No.12150351]

>>12150319
That's completely wrong.
You show that any map is homotopic to a constant map by using the retraction homotopy, and then you show that any two constant maps are homotopic by using the path connectedness.
>>12150343
FOCUS ANON IT'S NOT A LOOP THE ENDPOINTS AREN'T FIXED

[Anonymous No.12150355]

>>12150343
Yeah, I see now. how would I prove that I can choose paths such that [math]H[/math] is continuous. Or is there better way of proving this?

[Anonymous No.12150358]

>>12150319
You can't prove something that isn't true.

[Anonymous No.12150374]

>>12150355
Don't do it like that. Notice that your end points are not fixed, contract both paths to a constant path and then connect those points with a path.

>>12150322
<3

[Anonymous No.12150380]

>>12150374
>exactly identical to >>12150351
Great minds and all.

[Anonymous No.12150386]

>>12150380
Oops I missed that. Sorry for your loss if we are on the same wavelength.

[Anonymous No.12150398]

>>12149748
cope
>>12148061
funny that he had political appointment too ! interesteing

[Anonymous No.12150431]

>>12150343
>>12150358
It's true morons, he didn't say a based homotopy of paths. He's asking if two functions are homotopic. They obviously are.

[Anonymous No.12150456]

>>12150431
>>12150351
You both need to chill out. We obviously meant that the described function was not continuous in general.

[Anonymous No.12150469]

>>12150296
all of us make mistakes. including me, jacob lurie

[Anonymous No.12150481]

>>12150456
>We
Don't post for the other idiot.

[Anonymous No.12150540]

>>12150469
With all the HCT people literally worshiping him, I wouldn't be surprised if most of the category theory hate posts were actually sent by Lurie to vent out his disappointment and anger.

[Anonymous No.12150581]

>>12150431
BASED

[Anonymous No.12150582]

>>12150267
Functional analysis is important for certain things in geometry and probability (functional analysis pairs very well with measure theory).
Math of machine learning may be highly probabilistic.
The others are all great, but not essential for what you're doing.

[Anonymous No.12150632]

Anyone have a recommendation for a short introduction to PDEs? I've taken grad level analysis but have barely touched ODEs / PDEs. I've heard Evans is good but am hoping to find something more concise. Thanks <3

[Anonymous No.12150677]

>>12150632
Theory or computational?

[Anonymous No.12150678]

Anyone here experienced with [math]\LaTeX[/math]?

Let's say I want to make a table with where the first column is a series of numbers going from 1, 2, 3... all the way to 25
Now let's say that I wanted to add a row in between rows 3 and 4 so know the new row is row 4, but now I have to change the numbers from the old rows 4-25 to 5-26 to reflect this change. Rather than having to individually change each row number one by one, is there a way to assign some sort of variable to the cells in the column such that [math]\LaTeX[/math] assigns the nunbers to them during compilation?

[Anonymous No.12150685]

>>12150161
if f(x) > g(x) then f(a) > g(a) and F(a) > G(a) which follows from the linearity of the (anti) derivative.
Same with b. So F(a) - F(b) > G(a) - G(b).

[Anonymous No.12150728]

>>12150685
Even if this proof worked, it wouldn't work.
The errors are, in order:
[math]F[/math] and [math]G[/math] aren't functions, they're functions up to a constant.
Because of that [math]F(a) > G(a)[/math] isn't a statement that makes sense.
The entire thing above repeats for [math]b[/math].
Even if we had [math]F(a) > G(a)[/math] and [math]F(b) > G(b)[/math], we wouldn't have that [math]F(a) - F(b) > G(a) - G(b)[/math].

[Anonymous No.12150736]

>>12150728
I forgot the last and most important one:
The inequality you want to prove is actually [math]F(b) - F(a) > G(b) - G(a)[/math]

[Anonymous No.12150748]

>>12150685
>antiderivative
>assuming f and g are continuous
>implying any order relation would be preserved due to linearity
x maps to -x is also linear, if x > y then is -x > -y? dumbass

[Anonymous No.12150886]

>>12150677
Definitely theory.

[Anonymous No.12150997]

>>12150398
Is this the real Yukari?!

[Anonymous No.12151001]

>>12150161
We have that $f-g\geq 0\implies \int f-g \geq 0$. Since the upper integrals on the latter are a lower bound of the lower integrals of the former. $\int f-g = 0$ implies that $f-g=0 a.e.$. Thus, given that $f$ is strictly greater than $g$, we must have $\int f-g > 0$. Result follows.

[Anonymous No.12151012]

>>12151001
We have that [math]f-g\geq 0\implies \int f-g \geq 0[/math], since the upper integrals on the latter are a lower bound of the lower integrals of the former. [math]\int f-g = 0[/math] implies that [math]f-g=0 a.e.[.math]. Thus, given that [math]f[/math] is strictly greater than [math]g[/math], we must have [math]\int f-g > 0[/math]. Result follows.

[Anonymous No.12151019]

>>12151001
We have that [math]f-g\geq 0\implies \int f-g \geq 0[/math]. Since the upper integrals on the latter are a lower bound of the lower integrals of the former. [math]\int f-g = 0[/math] implies that [math]f-g=0 a.e.[/math]. Thus, given that [math]f$ is strictly greater than [math]g[/math], we must have [math]\int f-g > 0[/math]. Result follows.

[Anonymous No.12151074]

>>12150161
[math]\lim_{n \to \infty} \sum _{c=1} ^n [\frac{b-a}{n}(\textit{f} \ (a+\frac{b-a}{n}c))] > \lim_{n \to \infty} \sum _{c=1} ^n [\frac{b-a}{n}(\textit{g} \ (a+\frac{b-a}{n}c))] [/math]

[Anonymous No.12151077]

>>12151074
can I get better TeX limit and sigma suggestions please

[Anonymous No.12151092]

>>12150886
Evans' - Partial Differential Equations
https://drive.google.com/file/d/1LmwLdo7XebkjoIh5jybmpJ4wdW--WXLh/view

Working knowledge of measure theory is highly recommended but not necessary per se.

[Anonymous No.12151112]

>>12151092
He said in the first post he wanted something more concise than Evans you fucking retarded sperg
>>12150886
Try Arnold’s ODE’s and lectures on PDE’s. Both on libgen.

[Anonymous No.12151130]

>>12151112
Perfect–thanks!

>>12151092
Yeah, I was looking for something a little shorter. Lots of these grad level door stops work as references or with an instructor, but in my experience aren't great intros to a subject. Can't imagine learning algebra for the first time with lang, for example.

[Anonymous No.12151362]

>Get stuck on a proof in a textbook
>Feel like I have a simplification that is too good to be true
>After a while, decide to ask about in on MathOverflow
>Very nervous because I'm about to look stupids to other mathematicians
>Realise my mistake as I'm typing the question in as much details as possible

Thanks MO

[Anonymous No.12151368]

>>12150267
>Functional analysis
this goes with measure theory

also go directly for phd

[Anonymous No.12151448]

>>12151362
>pop into advisors office with a question
>go to the chalkboard
>start explaining the question
>realize the answer is easy halfway in
>look like an idiot

[Anonymous No.12151529]

>>12151448
Hence the superiority of MO over a human advisor.

[Anonymous No.12151541]

>>12148427
Does this have to do with 3d cube coordinates and shit?

[Anonymous No.12151556]

>>12151448
>>12151362
Yes, it's amazing how the moment you actually try to explain your problem to someone else FOR REAL, not just sketched, you quickly find your fault.

[Anonymous No.12151592]

>>12151368
he could be in europe (judging from the names of the courses and how he called it an msc he probably is). not so common to go straight to phd over there as far as i know

[Anonymous No.12151607]

>>12149785
Thanks anon! That was short and to the point

[Anonymous No.12151623]

>>12148427
3
If any mathemagician would not want coffee themselves, that mathemagician would know that not all of them want coffee and hence would have answered "no". So they all must want coffee.

Of course, the mathemagicians probably would be able to predict from their past experiences that all of them are likely to want coffee. They instead opt for this reply for a bit of fun and to test the intelligence of their waitress. I should try this sometimes when I'm out drinking with some colleagues.

[Anonymous No.12151645]

>>12149785
>imagine that someone comes up with a quick algorithm for travelling salesman problem in the special case when the cities are actually points on a plane, and the distances are just euclidean distances between the points.
Someone sort of did so recently. https://arxiv.org/abs/1807.06933 (quick is rather relative of course, Euclidean TSP is still NP-hard, but it is a nice result to get something ETH-tight in R^d)

[Anonymous No.12151856]

>>12151001
Works, you get a gold star.
>>12151074
I'm not seeing it, could you elaborate?

[Anonymous No.12151929]

Isn't measure theory a prerequisite for functional analysis? How can you take these side by side?

[Anonymous No.12151931]

>>12151929
By having the lecturer teaching measure theory speedrun it, while the lecturer teaching functional analysis start by concepts which don't need measure theory or explain stuff very roughly by saying "you'll see that in your measure theory course".

[Anonymous No.12151979]

>>12151929
As if you couldn't write down theorems about Fourier transforms with analysis and make notes that they generalize

[Anonymous No.12152009]

So the topology with the algebraic sets as closed sets is the Zariski topology right? But what if you take analytic sets as closed sets, how is it called?

[Anonymous No.12152020]

>>12151362
>>12151448
>>12151556
Isn't that why codemonkeys have the rubber duck method

[Anonymous No.12152089]

>>12152009
good question

[Anonymous No.12152110]

>>12151856
>Works, you get a gold star
No, it doesn't. He hasn't proved that the lower integrals are bounded below as he claims. This is nontrivial. Lower integrals take infs over segments of a partition, what if the function approaches zero on any open neighborhood? Then the lower integrals are all zero. He hasn't shown that such functions are not Riemann integrable. Also where is this "a.e." coming from, there's no measure.
Sort of shit I would give someone a 1/5 on if I were grading his problem set.

[Anonymous No.12152118]

>>12152009
Are uncountable intersections of analytic sets still analytic? Countable intersections are.

[Anonymous No.12152132]

>>12151929
You only really need the basic definitions and a few theorems. Dominated convergence, monotone convergence, Fubini and Tonelli will do. Also, this >>12151979 .

[Anonymous No.12152143]

>>12148251
By looking at remainders mod 9 I only manage to reach the conclusion that n = 3k (for example, 5^6 and 2^6 are equal to 1 modulo 9). Out of curiosity, for how large powers of n have you verified your hypothesis?

[Anonymous No.12152145]

>>12152020
Yeah, I remember hearing about it now that you mention it.
Maybe I should get myself some sort of rubber duck.

[Anonymous No.12152157]

>>12152020
>>12152145
I do the similar stuff to stuffed animals. I really recommend it, especially if you are to have a presentation of some sort. Good preparation.

[Anonymous No.12152173]

Life on the outside ain’t what it used to be

Lately I just shitpost on /mu/, /sci/, and /g/.

Well it’s the last thread I know. There’s only one place to go! I’m coming home, oh yeah, there’s /mg/!

[Anonymous No.12152228]

>>12152110
Oh right.
For some reason I read "f-g not a.e. zero", and my brain filled in "equate the Riemann integral with the Lebesgue integral, the Lebesgue integral with the L^1 norm and that one is non-zero because it's not a.e. zero."
In my defense it was seven in the morning, I was in a hurry and the broken Latex was confusing me.

[Anonymous No.12152248]

>>12152173
/mu/ is literally 18 year olds on average (and I mean it literally, for every 22yo you have two 16yo's)

[Anonymous No.12152311]

>>12152248
I really can't tell if anyone who goes on /mu/ actually plays an instrument. Literally filled with LARPers. At least here we have people commenting on what they're working on and it's harder to fake maths

[Anonymous No.12152338]

>>12152311
The only reasons to go to /mu/ are sharethreads and maybe /nbbmn/ in a few weeks.

[Anonymous No.12152458]

>>12152311
>It's harder to fake maths
>what is mathgen

[Anonymous No.12152467]

>>12148449
nevermind. I've figured it out

[Anonymous No.12152474]

>>12152467
No one gives a shit about your stupid fucking textbook.

[Anonymous No.12152519]

>>12152467
How did it go? Please give me the piece of mind.

[Anonymous No.12152540]

>>12150582
>>12151368

How much functional analysis do you think would be necessary? We get to do 2 courses on it, one a sequel of the other.

First one is about infinite hillbert spaces and operators on them, the second on infinite banach spaces.

How much do they apply to geometry and probability?

[Anonymous No.12152543]

>>12147642
>mfw

https://youtu.be/lDhKE2SKF08

>>12152338
>/nbbmn/
what's that

[Anonymous No.12152607]

>>12152519
You can construct N exactly so that it works. See the appendix of hatcher's algebraic topology

[Anonymous No.12152610]

>>12152474
the anime fag gives a shit

[Anonymous No.12152619]

my number sense is so bad. like, amazingly bad. has been ever since I've been a kid. I like topics like analysis and algebra, but anything number-theory related turns my brain into mush. I still don't understand division and congruence to this day.

[Anonymous No.12152626]

>>12152543
Nothing But Black Metal November.

>>12152607
Fine.

>>12152619
You will understand them. Even I do understand them, and I'm probably the most retarded person in these threads.

[Anonymous No.12152633]

>>12152110
>This is nontrivial.
maybe that is, I don't care, but it's also enough to upper bound the lower integrals of g by f. This is entirely trivial and leads to the same monotonicity statement I gave first.
>there's no measure
Lebesgue measure, and we know integrating over it is equivalent to the reimann integrals since f,g are Riemann integrable. If the integral of the difference [math]\int f-g d\mu[/math] is 0, then we have that the measure of any [math]\{f-g>1/n\}[/math] is 0. Thus since the difference is nonnegative, f-g a.e. and result follows.

[Anonymous No.12152642]

>>12152633
>>12152633
>Thus since the difference is nonnegative, f-g a.e. and result follows.
I mean that the difference is nonnegative and then this would imply f-g = 0 a.e. Therefore [math]\int f-g >0[/math]

[Anonymous No.12152670]

>>12152626
i still need to use my fingers to add and subtract

[Anonymous No.12152763]

>>12152458
Not just that, all you have to do is listen to enough grad students or professors talk about their work and memorize enough of what they said/wrote and you can convince a casual observer that you know what you're talking about. Of course you'll be found out if you actually try to do any math, but good con artists find a way to escape such situations.

[Anonymous No.12152773]

>>12152619
If you don't understand division and congruence you don't understand algebra.

[Anonymous No.12152799]

>>12152458
>>12152763

What I meant by doing maths was the ability, when given a theorem, to be able to prove theorems in said field you work in.

Sure mathgen makes up fake maths but to a mathematician I would've hoped they can tell you it's utter rubbish and in that why it is.

[Anonymous No.12152825]

>>12152799
>Sure mathgen makes up fake maths but to a mathematician I would've hoped they can tell you it's utter rubbish and in that why it is.

Yeah mathgen is obvious to any actual mathematician but it's possible for one mathematician not to be able to make heads or tails of something in another area. Math is more like a thousand different languages than one single language, and no one speaks them all.

[Anonymous No.12152841]

Anyone has any fun programming ideas involving advanced mathematics? Advanced like involving topology, algebra anything which sounds cool.

[Anonymous No.12152856]

>>12152841
Well there's always computational algebraic topology. Let's see those cohomology rings!

[Anonymous No.12152918]

>>12152841
Not exactly what you're looking for but writing a basic computer algebra system to me always seemed like a fun project. I'm sure you could also add some computational things in that suit your tastes.

[Anonymous No.12152959]

>>12151929
Understanding lebesgue dominated and fubinis theorem and how to use it can be done without have evet seeing the proof. This is the secret analysts don't want you to know.

[Anonymous No.12152961]

>>12152841
>>12152918
I'd be interested in doing something handson.

What's advanced to you and what do you like?
What do you want to get at.
I know some python, C++, Haskell, Idris

[Anonymous No.12152973]

>>12152841
I want a program that factors polynomials in finite finite rings through exhaustion.
>does it need to be exhaustion
Yes.

[Anonymous No.12153038]

>>12152973
Can't Sage do this?

[Anonymous No.12153044]

>>12152973
Sounds exhausting

[Anonymous No.12153047]

>>12152961
I quite enjoy things like analysis and probability but starting to develop a interest in geometry and topology. Don't have any particular result I want to arrive at mainly want to sharpen my programming skills with more maths based projects

Mainly program in c++ if that's any help

[Anonymous No.12153053]

>>12153038
Fuck if I know.
Chop chop lad, you could have implemented a class with finite field arithmetic by now.

[Anonymous No.12153059]

Has anyone mentioned that we have a NEW NIKOLAJ VIDEO BOYS AND GIRLS?

[Anonymous No.12153079]

>>12153059
Yes, I mean I posted it 20 posts above.
Although I won't do so if people are annoying about it, as in writing all cap posts.

>>12153047
Okay, although I still don't like the level.
Also do you want to do a project with people on github or just toy around on your own?

I really liked this paper, have a look if something is for you.
http://www.cs.ru.nl/~spitters/editorial.pdf

Or you like numerical analysis?

Do you see how to computably translate topology into coding? It's general tough.
The Heyting algebra stuff is highly relate to topology via frames (as in frames and locales), if you're into that
https://en.wikipedia.org/wiki/Pointless_topology

There's Lie theory libraries for C++, one could do something similar. I'm personally very interested in group interpolation, geodesic or otherwise, of two elements or more. There's papers on that.

[Anonymous No.12153082]

>>12153079
I still don't know the level*

[Anonymous No.12153179]

>>12152856
Computational algebraic topology sounds fun on paper, but is actually really shit and not worthwhile when you sit down to code it

[Anonymous No.12153207]

>>12152841
any kind of visualization tool is a 10/10 in my books.

[Anonymous No.12153211]

>>12153038
>announcing Sage
enjoy your ban

[Anonymous No.12153216]

>>12153179
We have different tastes I suppose.

[Anonymous No.12153276]

What is your favourite space-filling curve, /mg/?

[Anonymous No.12153310]

>>12153276
Experts believe your mom used to have curves before she filled space.

[Anonymous No.12153379]

>>12153276
your curves look a bit antisemitic, sir.

[Anonymous No.12153403]

I have a compact, connected topological space with a probability measure on its Borel algebra.
If I have a continuous function on this space, the intermediate value theorem guarantees that there is a probability measure concentrated on a single point such that the function's integral along this new measure equals the integral along the old measure.
Are there any ways of extending this to [math]n[/math] continuous functions and a probability measure concentrated on some [math]n[/math] points?

[Anonymous No.12153420]

>>12153403
Have you tried or are you just posting random stuff?

[Anonymous No.12153446]

>>12153420
Bit of both, bit of both.

[Anonymous No.12153469]

>>12153211
>>12153310
kek

[Anonymous No.12153607]

...how did they know? Are brazillians that powerful?

[Anonymous No.12153746]

>>12152619
Division
A into B equal sized groups
A into N groups containing B items each (containing fewer than B means fractional group)
Divide by -1 means groups become antigroups

[Anonymous No.12153749]

>>12147606
Who's this faggot

[Anonymous No.12153922]

>>12153749
Looks like Cédric Villani.

[Anonymous No.12154072]

>>12153079
Mainly just want to toy around on my own haha.

Lie theory and its implementation looks interesting to me, thanks for the other suggestions too bro

[Anonymous No.12154087]

>>12153749
some pathetic faggot who thinks being good with numbers entitles him to dress like a faggot in public

[Anonymous No.12154237]

Analyse: Is the triple venn model of representing set operations consistent with the non-graphic symbolic method?

>what are the conditions for being consistent?
Given arbitrary sets A B C and any operations on them, the resulting set must equal the set represented by the shaded region resulting from the same operations acting on the shaded arbitrary (elements are not pre-specified a certain subset within) circles A B C in the venn diagram

>are the conditions met?
Define a region to represent the set given by checking which circles it must be in/not be in to be in said set, which corresponds to checking which sets it must be in/not be in.

All operations act on two sets. Thus if we can show venn operations are consistent for all pairs of sets made from A B C, then chains of operations will be consistent as well

Assume union means the region enclosing both specified regions, and no regions outside the specified. Thus, the union of regions means to be inside, it must be in region 1 and/or 2. So it must be in set 1, and/or 2 - union.

Assume intersection means only the region that is in both specified regions. Thus it must be in region 1 and 2, so it must be in set 1 and 2 - it is in their intersection.

Assume subtraction of regions means a region, without any of the specified second region. Thus an element must be in region 1 without being in region 2, in set 1 without being in set 2 (the definition of subtraction is in 1 and not in 2)

>what if a certain subset of specific analysed set is null?
If a region refers to a set and that set is null, the region can be unencircled (the empty region). Assuming no regions are null (as in the general case) and then removing whatever happens to be null in the produced region (for a potential specific case) yields the correct set, as does pre-assuming a region is null before operating, left as an exercise to the reader

-

[Anonymous No.12154253]

>>12152311
I rarely go on /mu/ but I make music
https://gofile.io/d/mzpFjO

I went today to ask them if they knew an album like picrel but no one replied :{

[Anonymous No.12154258]

>>12154253
Nvm someone did reply finally

[Anonymous No.12154272]

>>12154087
Anon, you should kill yourself

[Anonymous No.12154310]

>>12152633
If you need to use the theory of the Lebesgue integral for your proof, you've been filtered. The point is to use basic facts on the Riemann integral.
>but it's also enough to upper bound the lower integrals of g by f. This is entirely trivial
No, it isn't. It isn't at all. Are you fucking stupid? You still haven't addressed the clear issue where the inf of f of f on any open interval might meet g.

[Anonymous No.12154315]

I'm going to run off into the dunkelheit thinking about void and topology. I leave you clamped faggots with some of the most god-tier thinking music known to mankind:

https://www.youtube.com/watch?v=Wdk5yuLvRCA

[Anonymous No.12154322]

>No, it isn't. It isn't at all. Are you fucking stupid?

[Anonymous No.12154360]

>>12154322
Based.

[Anonymous No.12154387]

>>12154322
>he gets stuck formally solving a problem that was explicitly called bait by the original poster and resorts to posting soijaks
Calm down lad.

[Anonymous No.12154523]

can space-filling curves have area? if they can, how?

[Anonymous No.12154709]

>>12154523
Sure, they have zero area. Any other questions Anon?

[Anonymous No.12154733]

>>12153607
For those wondering

[Anonymous No.12154737]

>>12147606
Group, Ring, Fields, Modules. Which of these have contributed to the legalization of LGBTQ rights the most?

[Anonymous No.12154748]

>>12154523
>>12154709
Define "área"

[Anonymous No.12154767]

>>12154748
Something something Lebesgue measure and something something Cartesian products.

[Anonymous No.12154771]

>>12154737
>which component of modern Algebra has sissified society more
Retarded question, no one element is more impactful than the other. they constitute a whole of effeminizing force.

[Anonymous No.12154775]

>>12154737
Rings, gay marriage was a huge step forward

[Anonymous No.12154793]

There is nothing wrong with tensors.

[Anonymous No.12154841]

>>12154315
You posted something other than Chopin. Is there a particular reason you claimed that it was "god-tier?"

[Anonymous No.12154844]

>>12154737
categories

[Anonymous No.12154858]

>>12154841
This but unironically.
https://www.youtube.com/watch?v=t-JD2bnNQvY

[Anonymous No.12155108]

We say RVs [math]X,Y[/math] are independent if their joint density equals the pointwise product of their marginal densities, [math]\rho_{X,Y} (x,y) = \rho_X (x) \rho_Y (y)[/math]. Is there anything interesting to say about the case where the [math]=[/math] is exchanged for either [math]\le[/math] or [math]\ge[/math]? (And the inequality holds for all [math]x,y[/math] of course)

[Anonymous No.12155111]

>>12154793
literally just a useless obfuscation of arrays

[Anonymous No.12155224]

>>12155108
>Is there anything interesting to say about the case where the = is exchanged for either ≤ or ≥?
No, because the difference is probabilistically irrelevant (call it f(x,y) - g(x,y), note that its integral over the entire sample space is 1 - 1 = 0, and apply to the coincidentally-supplied proof >>12151012).
Any reason you don't want to characterize independence in terms of [math]\sigma[/math]-algebras? Marginal densities have plenty of issues: for one, they need not always exist (e.g. the Cantor distribution).

[Anonymous No.12155229]

Algebraic geometry is based, but it's so goddamn frightening in a way.
>Let [math]X = V(I)[/math] be a closed subvariety of some affine space of dimension n over a field k
>Take [math]F_1,...,F_r[/math] a system of generators of I
>Let [math]x \in X(k)[/math] be a rational point.
>Consider the matrix [math]J_x = (\frac{\partial F_i}{\partial T_i}(x))_{1 \leq i \leq n,1 \leq j \leq n}[/math].
>Then X is regular at x iff [math]\text{rank} J_x = n - \dim \mathcal{O}_{X,x}[/math]
>Huh, that's neat, it really looks like good old differential subvariety stuff.
>Suddenly realize that a rational point just means a morphism [math]x : k[T_1,...,T_n]/I \rightarrow k[/math]
>I'm not even sure what [math]\frac{\partial F_i}{\partial T_i}(x)[/math] means

After five minutes of thinking, recollecting what I know, remember that Ker x is a prime ideal [math]\mathfrak{p}[/math] of [math]A = k[T_1,...,T_n]/I[/math] so I have to call [math]t_i[/math] the coset of [math]T_i[/math] in [math]A_{\mathfrak{p}}/\mathfrak{p} \cong k[/math] and then [math]\frac{\partial F_i}{\partial T_i}(x) = \frac{\partial F_i}{\partial T_i}(t_1,...,t_n)[/math] and things finally make sense.

Algebraic geometry is neat and fascinating, and in general algebra is definitely more pleasant that analysis to me, but I wish I had had the opportunity to do more abstract algebra in my first two undergrad years, when I started building my mathematical intuition.
Every now and then, I will read a theorem, an argument in a proof or even an exercise and panic for five minutes because I have no idea what's going on at all, before I can calm down, remember the facts and make sense of all of it. It's often not that complicated if I recall all the definitions I know, but it's hard to have them come naturally.

Does anyone share that feeling or am I just a brainlet?

[Anonymous No.12155231]

>>12155229
What's the scariest branch of math?

[Anonymous No.12155284]

>>12154310
>You still haven't addressed the clear issue where the inf of f of f on any open interval might meet g.
I don't care about that. g<f so lower integral of g is <= lower integral of f. that's enough for int f-g >= 0. and why not move to lesbegue if it makes a proof on this chinese forum a one liner. why do you want to spend so much time arguing about this boring shit in the first place.

[Anonymous No.12155290]

>>12155284
>>12154387
that wasn't me. not stuck, my proof is correct anyway.

[Anonymous No.12155292]

>>12153607
they know your career is over once you reach 24
hence the weird numbering

[Anonymous No.12155306]

You know how with some proofs you have to add a 0 to use some theorem, multiply by a 1 or define a function to get a neat direct proof?

Well, I think I'm a bit rusty after doing jack shit for 4 months straight. How do I get my powers back?

[Anonymous No.12155312]

>>12154737
Magmas

[Anonymous No.12155442]

>>12155312
This.

[Anonymous No.12155513]

>>12148427
1 because I "Aye" as in yes, Don't as in "No thanks" and Know as in "No".

[Anonymous No.12155518]

>>12155231
Derived Category Theory and/or any K-Theories

[Anonymous No.12155657]

I feel like there should be more ways to create a digital/cryptocurrency than the current blockchain one. Blockchains don't exactly feel elegant.
What's elegant is one coin being the solution to a difficult mathematical problem. This way one can mine coins by working on the problems, and perform transactions by giving others a solution. The validity of the coin can always be verified by checking that it in fact solves the problem.
There is only the double spending problem: I could give my solution to an arbitrary amount of people without them knowing, thereby using the one coin several times.
This is my approach to solve this: There is a large, decentralized collection of open problems/mathematical objects. This could be something like a list of a billion very large prime numbers. The list will be stored decentrally by the global collective of users and that's why every process has to be robust against a local version of the list not always being 100% up to date.
A coin is a solution to one of these problems, for example a factorization of a prime number. To mine a coin, I have to find such a solution through brute force. To give someone a coin, the following happens:
1) I give them my solution.
2) They take their solution and the corresponding problem and perform a certain mathematical operation which yields a new problem and also its solution.
3) They announce to the network that this problem has been solved and is now replaced by the new problem
4) The initial owner of the coin has now lost the value because the problem that it solved is now considered obsolete. The recipient of the coin now owns again one valid coin.

[Anonymous No.12155662]

>>12155657
Now this system still allows for double spending. One could send a coin to several people simultaneously and before the news have spread in to the network that the problem has already been solved, the transactions seem valid. But is this actually a problem? If it is sufficiently difficult and unlikely to perform such double spending, doesn't that just mean the value of a coin is now increased by some factor which is the expected number of succesful spendings. I feel like things could still work as a currency if everything is set up so the probabilities of inconsistencies average out to being insignificant on the large scale.

[Anonymous No.12155831]

>>12147606
Interesting challenge problem, should be able to solve using basic analysis concepts but would like to see someone use some more advanced things as well.

Let [math]A \oplus B = \{a \in A \hspace{2mm},\hspace{2mm}b \in B \hspace{2mm}:\hspace{2mm} a + b \}[/math]
Prove the [math]C \oplus C = [0,2][/math] where [math]C[/math] is the Cantor set.

[Anonymous No.12155916]

Dumb question but sqt is about to reach bump limit.

If [math]A,B_i[/math] are modules and [math]A\hookrightarrow \bigoplus_i B_i[/math] is an injection, does that imply that each [math]A\hookrightarrow B_i[/math] individually is an injection? Interested in both the finite and infinite index case. I know this is probably not true for infinite products which is why I'm asking.

[Anonymous No.12155924]

>>12155916
Not true, imagine mapping two points into two points with the same "x coordinate" but different "y coordinate"

[El Arcón No.12155931]

>>12155831
I can buy that. Very interesting.

[Anonymous No.12155955]

>>12152841
Sounds of math?
[math]y=\ln\left(x^{-2}\right)3\cos\left(x^{\frac{5}{3}}\right)\sin^{2}\left(3x\right)[/math]
Here is how it sounds:
https://vocaroo.com/i/s0n1BKDYUfNj

This one is even better:
[math]y=\ln\left(x^{-2}\right)3\cos\left(x^{\frac{5}{3}}\right)\sin^{2}\left(3x\right)[/math]
https://vocaroo.com/i/s0IfctxaMzBB
2 min but you need to listen to the entire tune, there are some interesting passages

This one is chewy:
https://vocaroo.com/i/s0qWhQEgJ92J

[Anonymous No.12155980]

>>12155229
Just to correct you, a [math]k[/math]-rational point is not just a ring morphism, but a [math]k[/math]-algebra morphism.
This is an important distinction for the following reason:

Suppose you have an algebraic variety [math]X[/math] over a field [math]k[/math], and let [math]K/k[/math] be a field extension. A morphism [math]\text{Spec }K\to X[/math] is equivalent to specifying a point [math]x[/math] of [math]X[/math] (obvious, since a field only has one prime ideal), and a field extension [math]K/k(x)[/math], where [math]k(x)[/math] is the residue field at a point. The latter comes from the fact that the morphism on the sheaves induce a local ring homomorphism [math]\mathcal O_{X,x}\to K[/math], ie the ideal [math]\mathfrak m_x[/math] corresponding to [math]x[/math] gets mapped to the zero ideal of the field [math]K[/math], so in particular it induces the field homomorphism [math]k(x)\to K[/math] (which is by definition a field extension).

In particular, if we take [math]K=k[/math] and further, that [math]\text{Spec }k\to X[/math] is a morphism of [math]k[/math]-schemes/varieties, then we force that the extension [math]k(x)\to k[/math] canonically takes [math]k[/math] to itself, so that [math]k(x)=k[/math]. And that's the definition of a [math]k[/math]-rational point.

Translating all this to the language of algebra, locally, the morphism [math]\text{Spec }K\to X[/math] looks like [math]k[T_1,...,T_n]/I\to K[/math], and the kernel of this morphism is a maximal ideal. Forcing the morphism to be that of a [math]k[/math]-algebra, then the kernel is a maximal ideal corresponding to points [math](a_1,...,a_n)\in K^n[/math] that are a solution over [math]K[/math] to the polynomial equations defining [math]I[/math]. Taking [math]K=k[/math] in particular gives us exactly one point for every such morphism.

Therefore, the [math]k[/math]-rational points are in bijection with the solutions to your polynomial equations.

[Anonymous No.12156105]

>>12155312
Why? They're not even an active area of research.

[Anonymous No.12156138]

>>12156105
It is underneath every (reasonable) algebraic structure.

[Anonymous No.12156150]

>>12156138
If they were studied in their own right would they be called lavas?

[Anonymous No.12156172]

>>12155931
Yeah! And honestly the proof is very pretty

[Anonymous No.12156177]

>>12156150
Sounds reasonable. It would have to be either of those anyway, because they provide an example of how commutativity doesn't imply associativity. This is done by considering the rock-paper-scissors game, and hence it has to be something geological.

[Anonymous No.12156181]

>>12155955
>Sounds of math
>What are fourier series

[Anonymous No.12156228]

>>12156181
yes, your point?

[Anonymous No.12156349]

>>12156177
>algebra of rock-paper-scissors
Reminds me of this: https://arxiv.org/pdf/1903.07252

[Anonymous No.12156421]

>>12155831
We know that any number in [0,1] has a ternary expansion, and if that ternary expansion only contains 0s and 2s, that number is in the cantor set.

With a given [math]x\in[0,1], x=\sum_{k\in\mathbb{N}} a_k3^{-k}\text{ with ]a_k\in{0,1,2}[/math].

Let b_k = 0 if a_k = 0, b_k = 2 otherwise. Let c_k = 2 if a_k=2, 0 otherwise.

Then (b_k+c_k)/2=a_k.

Thus we have the numbers [math]c_1=\sum_{k\in\mathbb{N}} b_k3^{-k}, c_2=\sum_{k\in\mathbb{N}} c_k3^{-k}[/math] in the cantor set, such that c_1+c_2=2x. Result follows.

[Anonymous No.12156458]

>>12155916
Hint: Generally, it is the case that if you have two maps f, g and gf is injective, then f is injective

[Anonymous No.12156464]

>>12155916
What if [math]A = \oplus_i B_i[/math] tho?

[Anonymous No.12156476]

>>12155224
You're absolutely right anon, thank you

[Anonymous No.12156510]

>>12155916
A concrete example: [math]A = \mathbb{Z} = B_1, B_2 = \mathbb{Z}/3\mathbb{Z}[/math]. No injection [math]A \to B_2[/math].

>>12156349
Hah! I would assume this is a manifestation of being based.

[Anonymous No.12156513]

Proofs are a meme.

[Anonymous No.12156566]

Thanks injection bros, turns out i didnt need it (I too thought if it were true it would be too powerful, which is why I had to ask)

[Anonymous No.12156574]

>>12156421
>[eqn]x\in[0,1], x=\sum_{k \in \mathbb{N}} a_k 3^{-k} \text{ with ]a_k\in{0,1,2}[/eqn]

What do you mean by [math]]a_k[/math]?

[Anonymous No.12156577]

>>12156574
Ok didn't realize your text needed fixing

>[eqn]x\in[0,1], x=\sum_{k\in\mathbb{N}} a_k3^{-k}\text{ with ]a_k\in\{0,1,2\}}[/eqn]

[Anonymous No.12156623]

>>12156577
yeah sorry I fucked that up haha. I'm so used to overleaf to check my bugs that I'm horrible at getting latex right the first time. The a_k are the coefficients in the ternary expansion of an arbitrary x in the closed unit interval.

[Anonymous No.12156687]

>>12156623
>Overleaf
I heavily suggest you compile natively, Tex studio is absolutely fantastic and can be coupled with language tool for an incredibly streamlined document writing experience

Yeah I know what the a_k coefficients mean, what I'm wondering about is what is that ] right before the a_k, is that a typo?

[Anonymous No.12156700]

>>12156687
Not him, but that was definitely a mistype trying to close the \text

[Anonymous No.12156825]

>>12156687
I really wish Overleaf would implement a user dictionary. Spellcheck is basically useless when it only has everyday English words.

[Anonymous No.12156855]

>>12156700
This
>>12156687
>I heavily suggest you compile natively, Tex
Yeah I know I should. I've gone through different workflows, but since I'm only typing up problem sets at the moment, I don't think it matters too much either way.

[Anonymous No.12156856]

>>12156825
Yeah, a [math]\LaTeX [/math] editor should probably have "isomorphism" in its dictionary. I mean, you can add things to your dictionary, but it should be in there by default

[Anonymous No.12156900]

>>12156856
What about 'homeomorphism'?

[Anonymous No.12156901]

>>12156856
It frankly is bizarre. One wouldn't think it would be too difficult to just scrape a few hundred math texts and arrive at a reasonable domain specific lexicon to add to the standard dictionary.

[Anonymous No.12156996]

>>12156458
Ignore this. I misunderstood the question

[Anonymous No.12157020]

>>12156700
My thoughts exactly, just wanted to be sure.

>>12156855
Yeah, the installation is incredibly fast and easy and it makes everything so much better you won't believe it.

>>12156825
Overleaf Is.fine if you're doing simple stuff but you would write a thesis there

[Anonymous No.12157113]

>>12154858
I was speaking unironically. Also, good choice of Nocturne. If you'd posted Op. 9 No. 2 I would have laughed at you.
Best Nocturne is Op. 27 No. 2.
https://www.youtube.com/watch?v=WJ8RVjm49hE

[Anonymous No.12157116]

>>12157113
Do you like Debussy?

[Anonymous No.12157120]

>>12155111
Tensors come equipped with a product structure (more specifically, a monoidal category structure) which gives them far more structure than an "array"

[Anonymous No.12157125]

>>12155284
>>12155290
you're never going to succeed in mathematics, you simply do not appreciate the spirit of it to even the most minimal degree.
>I don't care about that
never, ever touch analysis again

[Anonymous No.12157140]

>>12157116
Chopin is my second favorite composer, Debussy is my absolute favorite.
For me, it's L'isle Joyeuse (here played 11 times in varying styles):
https://www.youtube.com/watch?v=OrGMdjmgUPo

[Anonymous No.12157150]

>>12157140
My man. For me, it's La Fille aux Cheveux de Lin.
https://www.youtube.com/watch?v=gy65UdvuHYk

[Anonymous No.12157191]

>>12154841
Because it's god-tier
Chopin is depressing and kind of gay in my opinion

[Anonymous No.12157207]

https://www.chebfun.org/examples/stats/Smoothies.html

[Anonymous No.12157222]

>>12157150
Oh, that's a very nice piece.
In terms of a slightly longer, more involved suite I like the Images.
https://www.youtube.com/watch?v=L47SRue0gt8
In terms of orchestral works, I definitely love La Mer (though I know most people's intrigue with Debussy stays near the more calm solo piano stuff).
https://www.youtube.com/watch?v=FOCucJw7iT8
It's funny because many credit Debussy with being one of the founders of the impressionist music movement, and the imagery and color in his pieces are what draw me to them. But Debussy himself despised impressionism and claimed that his pieces were intended as ungrounded and romantic. Despite naming his pieces after real things which they are clearly meant to channel. It's very strange.
Indeed, his philosophy was to write music which runs in parallel to, say, a vision of the sea or a woman with flaxen hair, as opposed to music which is an imitation of such images. This philosophy is what I think made him capable of writing such transcendent music.

[Anonymous No.12157234]

>>12157191
Your music is just noise and electric guitars and other things like that. That's what I find depressing.
I do think that to be a good mathematician, or at least good in certain fields, is very much correlated with the kinds of music one prefers. I think this romantic/impressionist stuff I like is very reminiscent of why an analyst loves analysis. There's a very fine line between the chaotic/untameable and the elegant in analysis and in this kind of music, and constantly inching ones toes over the line (or perhaps being violently thrown back and forth across the line) is valuable to me in both math and music.

[Anonymous No.12157237]

>>12157222
Music has the right to children

[Anonymous No.12157248]

>>12157234
Did you actually listen to it though? The first minute is warbling noise but it coalesces into a coherent heavy grunge riff from the chaos, that's a beautiful intro imo. The name of the album is derived from an HP Lovecraft quote that goes "the sciences each straining in their own direction... piece together patches of licht until they open up a blinding vista and push us into a new dark age"

Once you're into the brunt of the album, the grungy repetitive riffs serve to soothe and not distract, and give thought a simple bass-ic tempo. This makes thinking easier. Classical music is beautiful but it distracts from actually thinking. Chopin specifically is just very depressing in tone.

[Anonymous No.12157254]

>>12157248
Jesus, if I wanted to read fucking /mu/ posts I'd fucking go there. What drivel.

[Anonymous No.12157273]

>>12157254
Anon, why are you being retarded on purpose? It's simple neuroscience, brainwaves prefer a simple pattern to follow or else they get scattered. If you get something simplistic with enough dopamine to continue the energy, you can easily fall into a deep trance. Minor variations are beneficial to keep dopamine and awareness, which is what my music had. Btw you're talking about music so don't complain when people reply taking about music. Maybe it is you who are drivel, drivel deep in the soul!

[Anonymous No.12157276]

Since we're already off-topic, do any of you lads like vilainness manga?
I've just read "I’ve Become the Villainous Empress of a Novel", and I must say I like it quite a bit.

[Anonymous No.12157312]

And I thought I was a bad poster... Anyhow, this could be of interest to someone of you analysis people:
>On the extension and kernels of signed bimeasures and their role in stochastic integration
https://arxiv.org/pdf/2009.10657.pdf
And this for us more algebraically inclined:
>Schur--Weyl duality over commutative rings
https://arxiv.org/pdf/2009.10166.pdf

[Anonymous No.12157326]

>>12157312
>And I thought I was a bad poster...
Oh, I wouldn't worry about that.

[Anonymous No.12157335]

at least the guys who posted street fighter winked at each other and fucked off

[Anonymous No.12157392]

>>12157312
Am I supposed to take that as you not liking villainess manga?

[Anonymous No.12157442]

>>12157326
At least not the worst.

>>12157392
I don't like it in a non-classical logic way, as I have not read it. That doesn't imply I would dislike it either.

[Anonymous No.12157448]

Are there any cute female mathematicians?

[Anonymous No.12157449]

Is there a good way to visualize basic statistics , or is it just a grind course where you learn all the formulas

[Anonymous No.12157455]

>>12157448
>(You)

[Anonymous No.12157473]

>>12157125
lmao

sometimes I forget that the intersection of math and 4chan contains some truly exceptional autism

[Anonymous No.12157476]

>>12157448
Define 'cute', define 'female', define 'mathematician'.

[Anonymous No.12157487]

What's the deal with clopen sets? Why is closed defined as "complement is open" rather than "contains boundaries"?

[Anonymous No.12157489]

>>12157487
>clopen
Another reminder that mathematicians should not be responsible for naming anything.

[Anonymous No.12157494]

>>12154315
holy shit. I got stoned out of my mind today and all I could think about was how I'll get home and listen to this record. and then I find it posted in a fucking /mg/ thread. here's (you) my fellow marijuanaut. don't listen to this blue pill >>12157234 here, stoner/sludge/doom is the ultimate transcedental experience. absolute god tier, nothing comes even close.

[Anonymous No.12157500]

>>12157494
>dudeweed
Please kill yourself.

[Anonymous No.12157504]

>>12157500
Why?

[Anonymous No.12157510]

>>12157494
Hey fellow weedian
Nice to have a comrade on /mg/
I don't smoke anymore cause I got psychosis but I was stoned for 2 years straight, it's what got me into math

>>12157500
>Please kill yourself
Anon, you're super duper gay....

[Anonymous No.12157525]

>>12157510
>it's what got me into math
me too. now I'm doing phd in topology.

>>12157500
>2020
>offended by drugs

[Anonymous No.12157528]

>>12154858
Love this piece. Played piano growing up and this was the first thing I learned that I actually really liked.

[Anonymous No.12157531]

>>12157487
Anon, given any topological space X and any subset A of X, one can divide X into three disjoint subsets: the interior, exterior and boundary of A. The interior and exterior are open, and the union of either of them with the boundary of A will be its closure. Furthermore, the closure of the interior of A is the same as the closure of A. Now, A is closed iff it is its closure (i.e. contains its boundary) iff its complement is its exterior. The importance of clopen sets comes when you are dealing with connectedness. You are interested in the possibility of your space having a clopen set that is neither the space itself nor empty.

[Anonymous No.12157565]

>>12157504
>>12157510
>>12157525
Kill yourselves nigger degenerates.

[Anonymous No.12157569]

>>12157565
How did you drink the koolaid that badly lol
>t. nonsmoker

[Anonymous No.12157576]

>>12157565
You need to be 18+ to post here.

[Anonymous No.12157599]

Does stats get more interesting when measure theory is involved or does it always suck?

[Anonymous No.12157635]

can you imagine being such a fucking moron you think you're welcome in the math general as a druggie? there are multiple containment boards AND multiple containment threads on this board for you. if you're going to waste our time here, at the very least pretend to be a well-adjusted adult. marijuana is disgusting and smells awful. imagine spending your life thinking you're superior to people who don't need a crutch to get through the day cheerfully.

[Anonymous No.12157639]

>>12157487
You should look into the Kuratowski Closure Axioms, a less common but equivalent way of defining topologies.

[Anonymous No.12157657]

>>12157639
There are other ways?

[Anonymous No.12157661]

>>12157657
Yes, look into the Kuratowski Closure Axioms, a less common but equivalent way of defining topologies.

[Anonymous No.12157664]

>>12157657
Pierdole kurwa https://en.wikipedia.org/wiki/Kuratowski_closure_axioms

[Anonymous No.12157685]

>>12157635
keep dreaming kid

[Anonymous No.12158339]

It's been over 5 hours without a post.

[Anonymous No.12158548]

Anyone knows stuff about Arakelov Geometry?
I'm looking for a theme for my research (starting PhD), and thought it looks interresting enough and probably still full of unsolved problems.
Am currently working my way through Liu's Algebraic Geometry and Arithmetic Curves. Thinking to pick up Soulé et al.'s Lectures in Arakelov Geometry.
Any thoughts on either the book or the field? Should I read Fulton first?