/mg/ maths general

[Anonymous No.12182535]

Artisanal graph edition
Talk maths
Formerly >>12176144

[Anonymous No.12182555]

>>12182535
Claim your turf, /mg/.

[Anonymous No.12182593]

Fuck Nikolaj and fuck the algebra whore

[Anonymous No.12182599]

>>12182555
Checking those trips from the coast.

[Anonymous No.12182604]

Kiss Nikolaj and kiss the algebra whore.

Anyway, is there anything about Viete formulas in C[x1,...,xn]?

[Anonymous No.12182617]

>>12182555
Nice trips

[Anonymous No.12182697]

>>12182604
several variables didnt exist until the late 1800s

[Anonymous No.12182817]

>a lift of a deformation retraction of B to A, starting with the identity map of E, is a deformation retraction of E onto [math]p^{-1}(A) [/math]. This lift always exists using the usual definition of a fibration
I don't get it. How does this work? pic rel

[Anonymous No.12182870]

>>12182555
Where's graph theory and combinatorics?

[El Arcón No.12182892]

>>12182535
>Artisanal graph

[Anonymous No.12182902]

>>12182870
Graph theory is computer science and combinatorics is arithmetic

[Anonymous No.12182912]

>>12182870
Graph theory is the slums of topology.

[Anonymous No.12182924]

>>12182817
https://mathoverflow.net/questions/178509/in-a-fibration-can-a-deformation-retraction-of-the-base-be-lifted-to-the-total
link for context

[Anonymous No.12182942]

>>12182912
CW complexes are basically finite digraphs so maybe you should watch your fucking mouth. :)

[Anonymous No.12182987]

>>12182593
Based

[Anonymous No.12183027]

Can someone give me a hand figuring out the solution.

I know to use separation of variables, then set both to =K. The main thing throwing me is the omega squared

[Anonymous No.12183044]

>>12182555
ummm i'm somewhere between "desert of dynamical systems" and "niggerland of functional analysis"

[Anonymous No.12183064]

>>12182817
Do you know the way cofibrations, fibrations and weak equivalences are related? If you have what is called a model category, you have three classes of morphisms, cof, fib and we. If you specify your weak equivalences somehow, then it is sufficient to specify just the fibrations or the cofibrations, as they will satisfy the lifting property where you have a square in which the vertical arrows are of interest.
(1) If the left vertical arrow is a cofibration, then then a lift exists iff the right arrow is both a fibration and a weak equivalence.
(2) If the left vertical arrow is both a cofibration and a weak equivalence, then a lift exists iff the right vertical arrow is a fibration.
(3) If the right vertical arrow is a fibration, then a lift exists iff the left vertical arrow is both a cofibration and a weak equivalence.
(4) If the right vertical arrow is a fibration and a weak equivalence, then a lift exists iff the left vertical arrow is a cofibration.
Weak equivalences in this case are those maps that induce isomorphisms for homotopy objects, groups for n>0 and the sets for n=0. This gives us a model structure (with some assumptions, irrelevant now). Now, note that the left vertical arrow is both a cofibration and a weak equivalence, and apply what I had as the second thing on the list.

>>12182555
Woods.

[Anonymous No.12183068]

Professor logged into zoom meeting with two computers so there was massive feedback the whole lecture
-_-

[Anonymous No.12183088]

>>12183068
good for him, he didn't have to listen to your tranny porn

[Anonymous No.12183092]

I suck at math. Mostly because I often mess up orders of priority, forget rules or just sloppily write down wrong shit.
How do I desuckify myself on this subject?
Are there games or something to slowly but steadily improve to get to a higher level of understanding so I regularly get my dopamine shots?

[Anonymous No.12183108]

>>12183088
Best thing about corona was my 70+ year old prof asking how to only share one window because she didn't want her porn to show up by mistake.

[Anonymous No.12183122]

>>12183027
try exp(f(x,t)) and see where it gets you

[Anonymous No.12183125]

>>12183108
What can you greentext this

[Anonymous No.12183126]

>>12183092
Take notes on plain notebooks, use pens and don't erase your errors, cross them off.

[Anonymous No.12183131]

>>12183064
is there a more elementary way?

[Anonymous No.12183141]

>>12183126
Where do I get examples to practice with that also show how to correctly solve things in case I'm too retarded to do it alone?

[Anonymous No.12183148]

>>12183122
I've been told to use T(t) = exp(iwt) and X(x) =exp(-ikt).
My initial speration of variables was U=XT.
I'm still getting stuck with the constants of differentiation for each of X and T. I'm don't know what I'm supposed to be getting for them and then what to do after exactly

[Anonymous No.12183156]

>>12183125
>be me
>in online class because of corona
>70+ year old prof asking how to only share one window because she didn't want her porn to show up by mistake
>mfw

[Anonymous No.12183159]

>>12182555
I inhabit a hut in the forests between the republic of probabilistan and the plains of analysis

[Anonymous No.12183169]

>>12183148
Again, let u=exp(f(x,t)), plug it in, on paper, not just in your brain, and see where you end up with.

hint: it will be another differential equation in terms of f that looks simpler.
don't be lazy

[Anonymous No.12183203]

What was Groethendick's 'grand vision' (my words) that he was working towards that his phd students opted not to pursue.

[Anonymous No.12183217]

>>12183169
I'm not being lazy. I'm required to do it a very specific way which is why I'm getting confused

[Anonymous No.12183223]

NEW NIKOLAJ VIDEO
>>12183203
They didn't see the might of the notion of topos. He saw them as the most unifying notion in mathematics.

[Anonymous No.12183229]

>>12182555
GET OUT OF MY SWAMP

[Anonymous No.12183246]

>>12183027
I quickly scribbled this

[math] u(x,t) = \frac{1}{2\pi} \int_R \phi_0(k)e^{ikx-i\sqrt{k^2+\omega_0^2}t}dk [/math]

any idea if this is anywhere near your solution

[Anonymous No.12183297]

>>12183044
based fields, but why the racism?

[Anonymous No.12183320]

So what journals do you guys read

[Anonymous No.12183339]

>>12183320
Arxiv, libgen, scihub, /mg/

[Anonymous No.12183340]

>>12183246
Where did you get the 1/2pi from?

[Anonymous No.12183351]

>>12183340
Fourier transform convention

[Anonymous No.12183355]

>>12183340
This was once revealed to me in a dream.

[Anonymous No.12183359]

>>12183320
I have a pile of unpublished xeroxed papers I'm reading through. Old people gave them to me.

[Anonymous No.12183366]

>>12183320
none
if you want to read some kind of regular publication pertinent to your interests, i recommend browsing arxiv and mathoverflow with tags of your choice

[Anonymous No.12183377]

>>12183351
Ah we haven't learnt Fourier yet, we're specifically supposed to do it by separation of variables, setting both sides equal to constant K. Then solve both ODEs, then eliminate K and end up with "w(k)".
As you can see here >>12183217
It's quite specific in the methodology.

[Anonymous No.12183379]

>>12182892
you cant call your graph artisanal, it looks like fat steaming artifical bullshit

[Anonymous No.12183387]

>>12183064
leave it to the anime tranny to take elementary alg top and then ignore the fucking question to explain nearly unrelated shit about categories
why dont you just explain what the alg top looks like next time you dumb retard

[Anonymous No.12183388]

>>12183131
Surely there is, I just can't think of anything at the moment. That's basically the idea anyway. You have an acyclic cofibration (meaning it is also a weak equivalence) and any fibration, and those guarantee the existence of your lift. They are using the model structure where cof is the class of all closed cofibrations and fib is the class of all (Serre) fibrations, by the way. Sorry, I'm almost falling asleep at the moment. Maybe you could check the stuff Mark Grant cited.

[Anonymous No.12183396]

>>12183387
Why don't you?

[Anonymous No.12183399]

>>12183387
I think this is excessively cruel, but I do also get really annoyed when people spout category theory definitions and can't give a single fucking example of what they're mindlessly repeating.

[Anonymous No.12183450]

>>12183399
I don't think it is excessively cruel at all. What I provided was an answer, but a bit bad one. The proofs for the 4 claims I presented can be found in the book by the guy who answered the question.

[Anonymous No.12183461]

>>12183377
I don't really get what your prof wants but maybe this is close: let

[math]X_{xx}=(E+\omega_0^2)X[/math]

set [math](E+\omega_0^2)=-\omega^2[/math] you get

[math]X(x)=c_1e^{i\omega x}+c_2e^{-i\omega x}[/math]

Second one is

[math]T_{tt}=ET[/math]

So I just get

[math]T(t)=c_3e^{\sqrt{E}t}+c_4e^{-\sqrt{E}t}=c_3e^{i\sqrt{\omega^2+\omega_0^2}t}+c_4e^{-\i\sqrt{\omega^2+\omega_0^2}t}[/math]

[El Arcón No.12183463]

>>12183379
I can do it. My graph is artisanal.

[Anonymous No.12183467]

>>12183461
is there any way to preview the latex because I hate this

[Anonymous No.12183468]

Speaking of Category theory, this is Graciela Salicrup. She was mexican mathematician who worked on CT and solved multiple theorems and problems, which flew under almost everyone's radar because they weren't translated to english until some years later after her death. By that time her contributions were already old news but she worked on them years before anyone else.

[Anonymous No.12183475]

>>12183468
That's sad anon.

[Anonymous No.12183477]

>>12183467
yes, in the top right of the post window

[Anonymous No.12183478]

>>12183467

[Anonymous No.12183494]

>>12183477
thanks man

[Anonymous No.12183505]

>>12183461
This is the next part

[Anonymous No.12183512]

>>12183505
>>12183461
And that's the final part.

And yes, I find his questions quite often needlessly confusing

[Anonymous No.12183514]

>>12183505
Yeah I don't really get this part, because I probably have already done it here >>12183461. Probably your prof has some exact plug and chug method that he wants to see. You should call your classmates

[Anonymous No.12183515]

Why is discrete math topics so much harder than continuous ones?

[Anonymous No.12183525]

>>12183512
This one is here >>12183246 but there is some sign discrepancy so work on that
The 2pi is inside A(k) probably. Your prof really doesnt like to be clear

[Anonymous No.12183530]

>>12183515
how do you measure hard if you compete against the best?

maybe it feels that way because discrete math problem statements are easier to grasp and other math just has an entry barrier of having to learn a lot just to understand the problems

[Anonymous No.12183536]

>>12183512
>>12183514
His notes aren't any clearer. Is there any textbooks or resource on PDEs you guys would recommend? Something that doesn't assume you know a whole lot, I want to get a more complete understanding of this.

[Anonymous No.12183544]

>>12183512
jej, I read superstition at first

[Anonymous No.12183546]

>>12183536
I assume you're doing QM so just read the griffith, his treatment of relevant PDEs is very clear and you can extend the methods to a large class of problems (like yours here)

[Anonymous No.12183549]

>>12183515
because the basic objects are simple, so you get straight to real work
compare that to alg top, where you spend most of the semester defining the homologies and stuff

[Anonymous No.12183552]

>>12183468
I'm no feminist fag but yeah it's sad a lot of people here in UNAM don't know about her.

[Anonymous No.12183564]

>>12183546
I'll be doing QM next semester, but I'm also just doing pure math. I know griffith's Electromag book is good but it's reassuring to hear Griffiths QM books is also good too.

[Anonymous No.12183580]

I know this sounds normalfag as fuck but do you feel lonely when you can't discuss your niche problem to anyone you know?

[Anonymous No.12183581]

>>12183552
Realistically there many mathematicians out there that are completely unknown that probably deserve a bit of fame

[Anonymous No.12183584]

>>12183580
Yes

[Anonymous No.12183585]

>>12183580
google, in that sense, has been my only friend for years

[Anonymous No.12183598]

>>12183580
yes
especially since i realised not even my advisor really gets that shit

[Anonymous No.12183605]

>>12183131
Okay, so theorem 4 is what you want to use https://www.mscand.dk/article/view/10791/8812

>>12183366
Have you put yourself on the arxiv mailing list?

>>12183580
Yes. My supervisor started doing stuff and then decided that I will do the rest. Nobody has read the paper he wrote with a few friends of his because it hasn't been published after 2 or 3 years, so it is basically just the two of us but realistically just me.

[Anonymous No.12183620]

>>12183605
>the light insight is broken but I still work

[Anonymous No.12183626]

>>12183605
>the light inside is broken but I still work

[Anonymous No.12183642]

>>12183580
yes, it's why specialization sucks.

[Anonymous No.12183672]

>>12183580
of course, it's an inherent part of being a mathematician. one must decide if it's worth it

[Anonymous No.12183677]

>>12183672
>one must decide if it's worth it
...what if you decide that it's not?

[Anonymous No.12183679]

Is there any advantage the Riemann–Stieltjes integral has over the Lebesgue integral or was the former simply a precursor to the latter?

[Anonymous No.12183682]

>>12183679
those notions are orthogonal, you wouldn't ask if the gamma function is better than the factorial

[Anonymous No.12183694]

>>12183682
>if the gamma function is better than the factorial
But it is.

[Anonymous No.12183708]

How the fuck do you choose a topic for your masters thesis? For the Bachelors thesis our prof gave a list of topics and I picked some almost arbitrarily. I know I can learn the material. But it seems like you need to be far deeper into a topic for a masters thesis, no?

[Anonymous No.12183711]

>>12183708
>How the fuck do you choose a topic for your masters thesis?
Pick something you're interested in.

[Anonymous No.12183715]

>>12183694
The one requires a theory of transcendental functions and the other requires multiplication

[Anonymous No.12183765]

>>12183711
Bad advice. Master's thesis should be on something you've been learning for a long time AND you're interested in. If you pick something you superificially 'like', you risk delaying graduation because it will take you too much time to learn material and churn original research out.

[Anonymous No.12183773]

Ngl kings, power series for e is messing my ass up. I don't know why is confuses me so much, especially if it's something like e^x(x^2 - 1)

[Anonymous No.12183779]

https://science.ucalgary.ca/science-live-stream

[Anonymous No.12183795]

>>12182555
In the tundra of topology, drinking from the river between the fractal lakes and the ocean of logic

Btw it should be called "Geome-tree forest"

[Anonymous No.12183804]

>>12183297
kill yourself

[Anonymous No.12183821]

>>12183626
What else is there to do, though?

>>12183708
Find something interesting, then find a suitable supervisor and ask them if they would be interested in working with you on a topic related to your interests. Then formulate the thing together, or just go with your idea if you already have a concrete vision of what you want to do and the supervisor agrees with your plan.

[Anonymous No.12183826]

>>12181376
I want to study math, yes. Additional requirements on top of what, just the application itself? I am a US citizen, not EU.

[Anonymous No.12183837]

>>12183711
but how do you guys find something interesting? Are you supposed to start read papers etc.? If I take lectures in roughly 3 subjects that cover the basic stuff of advanced subjects (intro to stochastic analysis, intro to nonlienar PDE), then just this exposure doesn't seem like it would lead to finding an interest.

[Anonymous No.12183848]

>>12182555
Mt infinity actually looks like the mountain from my visions.

[Anonymous No.12183858]

>>12182555
The hills of differential geometry should be "variable gravity zone of diff geo" And the drawing of intersecting wormhole rivers and other oddities

[Anonymous No.12183873]

https://www.youtube.com/watch?v=oPp2mIv4CyA

https://de.wikipedia.org/wiki/Emanuel_Swedenborg

[Anonymous No.12183886]

>>12183873
Why Burzum and not for example Arckanum? He was Swedish and not Norwegian. Also, why does the German article call him a theosophist while other languages say he was a theologian? https://www.youtube.com/watch?v=YKtyOovTCQg

[Anonymous No.12183899]

>>12183297
niggerland isnt racism, grow up

[Anonymous No.12183906]

>>12183468
i wanna cum on her

[Anonymous No.12183908]

>>12183804
Why?

[Anonymous No.12183910]

>>12183886
Because:
>burzum is epic
>my post wasn't referencing nationality
>sweden is liberal
>i have never heard of arckanum

[Anonymous No.12183918]

>>12183886
Also I listened to it and that song wasn't good in my opinion. It was grunts and screech. But Erblicket Die Tochters Des Firmaments is a deep grunge with a riff in it.

[Anonymous No.12183940]

What do we think of Matlab? I have to use it for my Eng. Math course practicals

[Anonymous No.12183948]

>>12183940
Easy to use with lots of different capabilities. It's 'aight.

[Anonymous No.12183958]

>>12183948
I agree. I don't know why it gets so much hate.

[Anonymous No.12184054]

>>12183940
I only used it for my Numerical Analysis class. It was ok. The free alternatives are Octave, Maxima and other one, but none can match it.

[Anonymous No.12184072]

>>12183320
My school gives jstor access. I sometimes read old issues of annals of mathematics

[Anonymous No.12184078]

>>12183505
Unrelated but does anyone know what that font is called?

[Anonymous No.12184107]

>>12183468
Man, that sucks

[Anonymous No.12184108]

>>12183908
see >>12183899

[Anonymous No.12184299]

>>12183906
she was cute

[Anonymous No.12184541]

>>12182535
Yoooo this shit look kinda sexy, have any more?

[Anonymous No.12184549]

>>12182555
landfill of calculus
ODE waste incinerator

[Anonymous No.12184962]

I’m sad bros. It’s my last semester and am joining industry after graduating. The job pays quite well but I’m really going to miss studying mathematics. The worst part is that I know in a year or two I’ll have forgotten almost everything.

Taking a few grad courses rn and part of me wishes that I’d be continuing on to grad school. But I’m no genius so I’d probably end up joining industry in any case.

[Anonymous No.12184967]

>>12183605
>theorem 4
My question was about WEAK deformation retractions. You know, the case of which the guys in the MO thread say it's clear

[Anonymous No.12184994]

>>12184962
I know that feel.

[Anonymous No.12185036]

>>12184967
If I understood your question correctly, you were asking why does the lift exist. In the theorem you have a closed cofibration and weak equivalence on the left side and a fibration on the right side.

[Anonymous No.12185140]

How do you geniuses interpret 2D Discrete Fourier transform?

[Anonymous No.12185163]

>>12183203

category of motives

standard conjectures

[Anonymous No.12185211]

>>12185036
in the pic of the MO post allen hatcher says that you don't need the closed cofibration for the case
of a weak deformation retraction

[Anonymous No.12185225]

>>12185036
>>12185211
he also said that it is "clear" and you only need the homotopy lifting property. But I don't see how

[Anonymous No.12185228]

>>12183708
>choose an area you're interested and experienced in
>go to a prof that works in that area
>ask for topic

[Anonymous No.12185290]

>>12185211
>>12185225
I don't know. I give up. It probably has something to do with the fibration being Hurewicz (he says that it has the lifting property for all spaces), and then you wouldn't need closed cofibrations. I can't see it.

[Anonymous No.12185295]

>>12184541
Here's an artisanal topological illustration. More in George Francis' "A topological picturebook".

[Anonymous No.12185314]

>>12184962
I never knew what it means for people to "forget mathematics." I think that's a meme.

You can also do math on the side. Just don't know anybody up.

[Anonymous No.12185323]

So, the quintic can't be solved by radicals, but if you add bring radicals you can solve it. But I've heard there are algebraic numbers that can't be written with those either. Is there some finite set of algebraic functions that let you write all algebraic numbers? If not, what about adding a few special functions too?

[Anonymous No.12185518]

Are there any books similar to this?

https://www.springer.com/gp/book/9789811572609

[Anonymous No.12185581]

How the fuck does one fourier transform xe^(-x^2-iwx), my books don't tell me shit how you do one of these and integration by parts doesn't fucking work. Any help?

[Anonymous No.12185585]

>>12185581
use wolframaplha

[Anonymous No.12185588]

>>12185585
I can get the answer to the question through that, but it doesn't show me any steps and how it comes to that answer

[Anonymous No.12185591]

>>12185588
you can maybe reverse engineer

>>12185518
HOLi moli

[Anonymous No.12185624]

Alright I've got a choice between a module in Algebraic topology, Set theory and commutative algebra.

As someone who wants to research in probability in the future, which module would you consider to be the most beautiful /mg/ ?

[Anonymous No.12185692]

Why are Ideals called like that?

[Anonymous No.12185693]

>>12185581
well page 11-15 https://web.stanford.edu/class/ee102/lectures/fourtran

you get a special case of t^kf(t), so for you it is

j dF[f]/dw
with f[x] =e^(-x^2-iwx)

so find F[e^(-x^2-iwx)]

[Anonymous No.12185708]

>>12185692
Kummer imagined ideal factors of a ring element, but they didn't actually exist.

[Anonymous No.12185712]

>>12182555
Give me a good slice of the Hills of Differential Geometry

[Anonymous No.12185718]

>>12185323
Bring radicals aren't algebraic functions.

[Anonymous No.12185724]

>>12185718
The wikipedia page on algebraic functions gives them as an example https://en.wikipedia.org/wiki/Algebraic_function

[Anonymous No.12185735]

>>12185724
I see what you mean now. I would've only defined an algebraic function as the first class of examples given there, but that definition makes sense.

[Anonymous No.12185737]

>>12185624
Well AlgTop is certainly least useful for probability, and you won't need too much set theory either.

Just learn what a sigma algebra and a generating function is, maybe some of those courses help with that.

[Anonymous No.12185771]

Can someone help me with the following?

Prove that an increasing sequence {x_n} does not converge to any point in the lower limit topology (or Sorgenfrey line).

[Anonymous No.12185783]

>>12185737
I thought that both of them weren't as useful for probability but I would like to know what /mg/ opinions on which module would be the most enriching for a pure mathematician.

I've already picked the modules which would help set me up for prob, I just have a few courses left to spare so would rather put them in more pure fun subjects

[Anonymous No.12185796]

>>12185783
I think the "fun" of them is purely a matter of personality/taste.

>>12185771
Something about you approaching from the wrong side somehow?

[Anonymous No.12186013]

What does the exponential function actually do? I was taught that raising a number n to some power m was the same as m amount of n multiplied together, so 2^3 is three twos multiplied (2×2×2). However if you do something like 2^2.5, the result is not two and a half twos (2x2x1) but instead it is (2x2x(sqrt(2)). What exactly does it mean for something to be multiplied by itself half of a time?

[Anonymous No.12186031]

>>12186013
Square root

[Anonymous No.12186086]

>>12186031
That doesnt really answer the question, thats just writing it in a different way. And I even used the square root to show that I do already know that something raised to 1/2 is its square root, and I also know that something raised to 1/3 is the cube root and so on.
I guess another way i could ask is, if I had something totally random like 2^4.62279, how would I find the exact answer without using a calculator?

[Anonymous No.12186092]

https://www.youtube.com/watch?v=eJZTeiEzQjM
>the year is 2362
>the one world government has failed and warfare has ravaged the globe's infrastructure
>the population is a mere 800 million
>various regions are separated by military juntas and strongmen
>in the ruins of old Chicago...
>the seat of power lies in a brutalist bunker
>blaring music at 150 decibals to control the energy of the population
>the Black Elite rules the club
>imposing strict restrictions, taxes, and harsh raids on the neighbors
>controlling the drug flow to ensure subservience and power
>at the seat, is Jah the 13th
>53 years old, out of a possible lifespan of 340, a young leader
>he plots escape from this world, patronizing arts and science
>requisitioning materials and kidnapping the finest minds from neighboring regions
>as he sips tea, this song blares across the concrete matrix...

[Anonymous No.12186141]

>>12186092
>he laughs to himself
>"they were arrays of numbers all along"

[Anonymous No.12186143]

[Anonymous No.12186148]

>>12186013
2^2.5 is just an extension based on the notions that 2^a * 2^b = 2^(a+b) and that a fractional power is a square root, so 2^2.5 = 2^2 * 2^1/2, and 2^1/2 becomes rt 2. The reason why a^1/n is nth root of a is because:
>rt2
>2
>2rt2
>4
>4rt2
>8
or
>nthrt a
>nthrt a ^2
>...
>nthrt a ^ n = a
>a*nthrt a
>...
So doing a fractional power just breaks it down from whole steps to partial steps. Instead of 2^(n*1) being multiplying by full twos, you are doing 2^(n*1/frac) being multiplying by partial twos. Except the partial twos aren't fractions of 2, they are numbers that preserve the step size.

[Anonymous No.12186154]

>>12186141

[Anonymous No.12186185]

>>12186154

[Anonymous No.12186232]

>>12186013
If multiplication is just repeated addition, how can you multiply by 2.5?

[Anonymous No.12186235]

>>12186154
>>12186185

[Anonymous No.12186250]

>>12186235
Based.

[Anonymous No.12186269]

>>12186232
Because you'd add a 0.5

>>12186235
Proof?

[Anonymous No.12186283]

>>12186235
Was Teichmüller really an intuitionist?

[Anonymous No.12186296]

>>12183677
you stop

[Anonymous No.12186299]

>>12183940
garbage, use python + numpy + scipy
strictly better

[Anonymous No.12186312]

>>12186086
that is in its exact form
you can calculate a close rational by hand using power expansion

[Anonymous No.12186322]

>>12186013
that's not what it means. just because something has a simple definition on the integers does not mean that definition extends everywhere.
the exponential for complex numbers is defined to be the analytic continuation of the exponential for real numbers.
the exponential for real numbers is defined to be the unique continuous extension of the exponential for rational numbers (by uniform continuity on expanding bounded sets of rationals).
the exponential for rational numbers is defined to be the additive-to-multiplicative homomorphism which extends the exponential on integers and on reciprocals of integers to the rationals.
the exponential on reciprocals of integers is defined uniquely by existence of roots of positive numbers in the reals.
the exponential on integers is defined by this multiplication definition for positive integers, extended to 0, and then to the negative integers by declaring that one take the reciprocal.
your little intuition for the exponential only applies to the most base step, the exponential on positive integers.

[Anonymous No.12186341]

>>12186322
That's a bad take anon. The question is why do we extend from the base step in the ways that we do. And since we extend in natural ways, what do our extensions say about nature?
>the whole numbers are just arbitrary symbols, nothing to do with our physical universe!!

[Anonymous No.12186359]

>>12182555
>plains
Is the joke that analysis is boring?

[Anonymous No.12186367]

>>12186359
planes

[Anonymous No.12186584]

>>12186283
the joke is that both were nazis

[Anonymous No.12186603]

>>12186584
Brouwer too??

https://en.wikipedia.org/wiki/Mathematics_in_Nazi_Germany
>In the 1920s, Hilbert became involved in a dispute with L.E.J. Brouwer, a Dutch mathematician whose support for intuitionism had not been widely accepted by Germany's mathematical establishment.[3] Intuition (Anschauung) was contrasted with "modern abstract" mathematics like formalism.[1] There was a rivalry in those years between Berlin and Göttingen, and Berlin sided with Brouwer against Hilbert in the dispute.[4] The dispute took on an ideological dimension as Brouwer presented himself as a "champion of Aryan Germanness". When Brouwer objected to Ostjuden (German Jews of Eastern European descent) writing for the journal Mathematische Annalen, Hilbert removed Brouwer from his position as editor. The Nazis offered Brouwer a position at the University of Berlin in 1933, which he declined. Even so, the Dutch government suspended Brouwer in 1945 because of his connections to the Party; he was, however, eventually reinstated.[3]

Huh...

[Anonymous No.12186651]

Anons how do I learn math from the ground up? I'm either a total brainlet or at some point one of my math teachers didn't explain some of the core concepts right and since math builds upon previous knowledge then the rest of my life I couldn't understand almost all of the topics, anything I could do to fix this problem?

[Anonymous No.12186700]

>>12186651
Give an example and I will explain it

[Anonymous No.12186709]

>>12186603
How did they decide which math was aryan and which not? Why is formalism considered non aryan and intuition considered aryan? It makes sense to me intuitively but I want a formal explanation (due to my part jewish heriage)

[Anonymous No.12186743]

>>12182535
Guys what is this x-->e^x notation called? the book is artin's algebra and it does not even mention what this is.

What is this ??

[Anonymous No.12186747]

>>12186700
Everything from 4th grade and up is a solid start.

[Anonymous No.12186755]

>>12186743
it means: the function "exp" is defined by the formula exp(x)=e^x

[Anonymous No.12186761]

>>12186755
why did not the author write it that way? Id this some boomer stuff?
What is that symbol called? How to read it.

[Anonymous No.12186777]

>>12186761
i read it as "maps to"
the more common version of this symbol is [math]x \mapsto e^x[/math]

[Anonymous No.12186778]

>>12186747
What were you doing in 4th grade?

[Anonymous No.12186794]

>>12186709
An interesting source for this stuff is "Mathematicians under the Nazis" by Sanford Segal. Some quotes:

>In other words, “Deutsche Mathematik” was not simply a question of expelling Jewish professors from theuniversity or deemphasizing the roles of non-German nationalities in the creation of mathematics; it was the quite serious matter of discerning what was atypically “Nordic” mathematics suitable for and to the new German state and itsaspirations. This task also involved the historical problem of finding common Nordic elements in the great German non-Jewish mathematicians from Kepler to Hubert that could be shown as lacking among the Jews as well as the French.

>For the Nazi mathematician, the issue was the ability of mathematics to “relate to real objects,” where geometric configurations and integers were admitted as examples of “real objects.” The question was one of restraining mathematical abstractions, of permitting only that which was related to a concrete physical world. If “Deutsche Physik” were to be predominantly experimental, “Deutsche Mathematik” was to have some connection, however tenuous, with the mathematics of everyday: the integers, geometry, probability. The concept was vague (and needed to be so), and there is no use nor value in attempting to make it more precise.

[Anonymous No.12186795]

>>12186777
and i know some author says "send to". I was getting crazy over what are they sendin, what is sending anyway? It's like they played dart in their head and sends x to f(x).

[Anonymous No.12186830]

>>12186778
Definitely not paying that much attention and like I said the teacher was probably average because I remember everyone else having no clue about stuff too.

[Anonymous No.12186840]

>>12186830
I meant what was the subject category.

[Anonymous No.12186848]

>>12186794
Honestly, that's pretty cool. I would say my beliefs are as follows:
All math is natural. Some math is obviously natural, some not. No math should be rejected, and no math should be accepted without finding how it is natural. The goal of math is connecting to nature but constraining nature to the obvious is a weakness.

[Anonymous No.12186855]

>>12186794
An example of Aryan vs Jewish mathematics is in the definition of [math]\pi[/math]: Aryan mathematician Bieberbach defines it more concretely as the ratio of a circle's circumference to its diameter, the Jewish Landau defines more abstractly as twice the smallest positive zero of cosine, cos itself defined abstractly as a power series.

The journal Deutsche Mathematik published many kinds of content, student work, book reviews, research etc. Probably the most notable publication is Teichmüller's paper on extremal quasiconformal mappings, laying out Teichmüller theory

[Anonymous No.12186860]

>>12186848
>No math should be rejected, and no math should be accepted without finding how it is natural.
What does it mean to "reject" math, then?
Would you not permit someone do/publish they math they like, just because you don't see how it ties to "nature" or reality "naturally". Whatever that means.

>>12186795
Are you autistic?

[Anonymous No.12186884]

>>12186860
I wouldn't bar them from publishing, but I would encourage all formal math to be explained in intuitive terms before it is considered "well understood." I believe such an intuition always exists. Also, I do not simply mean geometric intuition. If there is a new realm of math that can't be visualized at all, that math should be understood in terms of its own natural symbol and operation set. The ways that the symbols can be manipulated create a new layer of abstraction and we can seek out how things we've already discovered like geometry, fit into that symbol network. For example, if there is something that is commutative, only in a symbolic sense, that's fine, because we can notice how commutativity occurs in physical areas. Then physics is a subset of a broader abstraction inherent to nature.

[Anonymous No.12186892]

>imagine thinking you're beyond philosophy when you've been fully cucked by platonic metaphysical incelibacy and are too autistic to question it.

[Anonymous No.12186926]

>>12186892
I don't even know what you're referring to anon. An intuition existing can be a nonplatonic notion if you replace it with "an intuition is generatable." I question it often but I believe in it because this is the pattern I notice in reality.

[Anonymous No.12187016]

On that note, there's 400 page book of collected philosphically minded articles on the topic
>One Hundred Years of Intuitionism (1907-2007)
https://www.springer.com/gp/book/9783764386528

To bring the discussion back to mathematics, but still stir the pot, here's a question.

What's the cardinality of the Cauchy real number [math]r=17[/math]. Smugpepe face.

[Anonymous No.12187021]

>>12186860
>Are you autistic?

[Anonymous No.12187035]

>>12186892
who are you quoting?

[Anonymous No.12187038]

>>12182535
Might be a stupid question, but how did people create graphs like in pic in those times? They seem more lively to me for some reason compared to modern ones.

[Anonymous No.12187047]

>>12187038
When I visited Göttingen, they had an age old clay version of it standing in a vitrine

[Anonymous No.12187049]

>>12185693
Thanks a lot

[Anonymous No.12187059]

>>12187049
You are quite welcome.

[Anonymous No.12187062]

>>12187038
what do you expect here? someone computed some points, and then used a ruler and pencil to make a fairly precise picture

[Anonymous No.12187063]

>>12187049
>>12187059
I love you bois

[Anonymous No.12187073]

>>12187062
They hand-drew every pic for books back in the days? I don't know, it just seems like there would be a better option than hand-calculating some points and draw over them. They look so comfy to me.

[Anonymous No.12187083]

>>12187073
>They hand-drew every pic for books back in the days?
How else would they have done it?

[Anonymous No.12187084]

>>12186840
I have holes on almost every subject category, do you actually want to explain all of that? I'm just looking for resources in which I can just go through on my own time.

[Anonymous No.12187097]

>>12187083
I don't know. I mean, computers were around when these kinds of images were still used in the literature.

[Anonymous No.12187117]

>>12182555
>Projective plain? Yeah, that's where I live, how could you tell?

[Anonymous No.12187127]

>>12186743
It's called bullshit notation that nobody uses for a good reason. Alternatively it might be called "human error" and should read [math]x\mapsto e^x[/math] instead.

[Anonymous No.12187131]

>>12187117
Based.

[Anonymous No.12187135]

>>12187127
relax dude
someone used a squiggly arrow instead of a straight arrow with an additional line
it's not a big deal

[Anonymous No.12187145]

>>12187135
>it's not a big deal
It kind of is.

[Anonymous No.12187154]

>>12187145
big deal would be writing "nigger" instead of [math]\mapsto[/math]

[Anonymous No.12187157]

>>12187154
Don't give me bad ideas anon...

[Anonymous No.12187309]

>>12187157
NIGGER MAPS GLOW IN THE DARK!!!

[Anonymous No.12187319]

>>12187309
I ONCE INTEGRATED A NIGGER MAP WITH MY CAR

[Anonymous No.12187402]

>linear algebra quiz today
>asked to take inverse of a 2x2 matrix
>"ok i'll just take the determinate then multiple by 1/det"
>take determinate
>it's 1
where did I fuck up

[Anonymous No.12187482]

>>12187402
swap entries on the diag, change sign on the counter diag
easier just to do row ops than ectually remember that garbo

[Anonymous No.12187491]

>>12187482
I ended up trying to do row operations and guessing what the inverse might be but I think I fucked it up.
This is why I fucking hate math professors man.
They teach you a technique and meme you into thinking it always works (just take determinate durr) then they give you a problem on a quiz or exam where they designed it so that their meme technique DOESN'T work just to FUCK you over. What a fucking prick
>haha this determinate is 1 try multiplying by that you little fuckers
what the fuck

[Anonymous No.12187496]

Hey lads. Does anyone else have genuine ADHD? As in, not caused by internet or distractions. It makes doing math 3x harder because concepts keep falling out of my brain. My brain is optimized to have vaguer thoughts and make arbitrary connections. When I do math sometimes I start thinking about other things. But part of the issue is also flow. I restrict my flow of thought by instinct, even when it goes on to different but related mathematical ideas or especially if it gives me an intuition to be explored that I'm not sure if its valid. I also have a large fixation on whether concepts are valid and how to generate other states of them which overloads me with data that I get lost in. If I am sleep deprived I often think clearer because of lacking inhibition. But my naturally subpar working memory (I started with 3-4 objects and have been training to to be higher), and my thought process that is more flowing and less discrete makes it hard. Any tips?

>>12187084
Yes, if you want to. I will write up intuitive explanations of simple math, as it is fun for me.

[Anonymous No.12187595]

>>12186341
>why do we extend from the base step in the ways that we do.
Too make a smooth curve, dummy.

[Anonymous No.12187680]

https://www.quantamagazine.org/building-the-mathematical-library-of-the-future-20201001/

[Anonymous No.12187700]

>>12187496
Well, then so it is good for both of us you can recommend me some good resources and then you can talk about some of the things I should watch out for, concepts that might be hard for me to get straight away and that type of stuff, and then you can explain them or talk about how I should approach it since you find it fun to do that. I would like something like that since it is the most efficient way for me at least.

[Anonymous No.12187717]

How do you decide which proofs to study / learn?

On a weekly basis I have at least 100 pages of reading to get through, and it's not possible to study each proof in detail while also finishing problem sets. But I struggle to find a middle ground between skimming and analyzing/verifying each line.

[Anonymous No.12187734]

>>12187491
ur missing a step or two there bro, try paying attention in class [eqn]\begin{pmatrix}a&b \\ c&d \end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b\\-c&a\end{pmatrix}=\frac{1}{\det(A)} \begin{pmatrix} d&-b\\-c&a\end{pmatrix} [/eqn]

[Anonymous No.12188012]

Read every book by Serre and you will be wiser

[Anonymous No.12188134]

>>12188012
This.

[Anonymous No.12188155]

Anyone care to explain to a fag a probability question? Ten fair dice are rolled, if order doesn't matter what is the total number of outcomes. Six choose ten doesn't work, so I considered getting the total number of outcomes subtracting the total number of duplicates results, however I don't know how to get the total # of duplicate results. Can't seem to wrap my head around it.

[Anonymous No.12188173]

What is a strongly Choquet space? Is the same as a Choquet game? Internet is giving me mixed answers.

[Anonymous No.12188175]

>>12187496
try not posting the first retarded self-concerned rambling nonsense that comes into your head in a thread for discussing mathematics, fucking faggot

[Anonymous No.12188356]

Reminder, Grothendeick has a jewish refugee, whose family was destroyed by the far right. Grothendieck personally supported non western refugees. Grothendieck is far smarter than you. Therefore you should support refuges no matter what region they come from.

[Anonymous No.12188396]

>>12188356
>Grothendieck is far smarter than you. Therefore you should support refuges no matter what region they come from.
Invalid argument, brainlet. Try again when you understand freshman formal logic.

[Anonymous No.12188427]

>>12188155
"six choose ten" sounds like 0 to me, there are 0 ways to choose 10 things if you only have 6 of them.
Think about it this way instead. You roll the dice and then you move them into piles based on their number. So how many ways are there to put 10 things into 6 piles (where some may be empty)? This is the same as order not mattering, because no matter which die has a 1, it has the effect of just another die joining the 1 pile.

Ok, if you need to know how to solve the auxilliary problem I posed, split the piles up with walls. You need 5 walls if the piles are all in a row. Now the problem is, how many ways are there to place 5 walls in between a row of 10 objects? The answer is the number of ways to put 5 things in 5 + 10 spots. So 15 choose 5.
(Equivalently, 15 choose 10, you can think of this like choosing where the 10 dice go and then putting the walls in the remaining 5 spots just as well as you can think of it like choosing where the 5 walls go and putting the dice in the 10 remaining spots).
this is weird shit, and it doesn't make any fucking sense, but that's what combinatorics is. maybe someone has a more natural description of this that requires less combinatorial bijections.

[Anonymous No.12188468]

>>12188396
one of the, if the not greatest mathematicians of the 20th century was a jewish refugee from the nazis. A subhuman in their ideology. Is it really unlogical to say that a greatest mathematician is waiting in the form of a south/CENTRAL American refugee fleeing to the united states or a Syrian refugee in Europe? Of course not. Oppose right wingers and support refuges, it is good for the mathematical and intellectual cultural of whatever country you are from

[Anonymous No.12188496]

>>12188468
>Greatest mathematician of the 20th century
>Not von Neumann

[Anonymous No.12188667]

>>12186848
>The goal of math is connecting to nature
I love how atheist rationalists always try to cast their mental gymnastics on to nature as some way to discriminate between good and bad mental ramblings. I love how their criterion for ''agreement with nature'' is completely build in their head and they do not even agree on what ''agreeing with nature'' means or not.

[Anonymous No.12188672]

>>12188356
>Grothendeick
He impregnated dozens

[Anonymous No.12189079]

My mind... it's expanding... from all the maths.

[Anonymous No.12189083]

>>12187016
>What's the cardinality of the Cauchy real number
Isn't the cardinality of every Cauchy real number the same?

[Anonymous No.12189102]

/mg/, I must confess, I’ve been doing “Elementary entry math” in Khan academy for hours upon hours in hopes of one day getting every single course 100%ed, is this small or big brain?

[Anonymous No.12189110]

>>12187402
>>12187491
are you implying there's something weird with multiplying by 1/det(A) if det(A) = 1 ?

[Anonymous No.12189113]

>>12189102
big brain; if you can add "100% on Khan Academy" to your resume you will guaranteed get any job you want, 300k starting, and it will impress all the women

[Anonymous No.12189123]

>>12189113
Thank you anon

[Anonymous No.12189124]

>>12189083
The joke is that the finite ordinal 17 has has cardinality [math]|17|=17[/math]
while the real 17, being some equivalence class of objects, each of type [math]{\mathbb Q}^{{\mathbb N}}[/math], is has cardinality

[math]|17|=|{\mathbb R}|[/math] or something.

So [math]{\mathbb R}[/math] holds [math]|{\mathbb R}|[/math] elements of size [math]|{\mathbb R}|[/math].

A converging Cauchy sequence of real numbers is...

The place in the von Neumann hierarchy where the Cauchy reals pop up is...

Pls fill out the blanks.

[Anonymous No.12189214]

Consider the series [math]\sum^\infty_{n=1}\frac{\tan(nx)}{f(n)}[/math]. For which functions f is true that this series converges for almost all x? When it does converge, is the sine of the sum integrable?

[Anonymous No.12189241]

Need notation advice.

I have a paper where I don't really use Cartesian products much except for one part. I want to talk about the Cartesian power of an element, e.g. [math]a[/math] to the Cartesian power [math]n[/math] would be [math](a, a, \ldots, a)[/math]. What's a good shorthand notation for this? Is it acceptable to write something like [math]a^{\times n}[/math], in analogy with how people write [math]a^{\otimes n}[/math]?

[Anonymous No.12189258]

>>12189241
Define that notation in the beginning of your paper. Then it will be ok to use that.

[Anonymous No.12189268]

>>12189124
Let [math]a\colon ([0,1]\times{\mathbb N})\to {\mathbb N}[/math] be be the position of the [math]n[/math]th one in the binary decimal expansion.
E.g.
[math]a(.10110110.., 1)=1[/math]
[math]a(.10110110.., 2)=3[/math]
[math]a(.10110110.., 3)=4[/math]
[math]a(.10110110.., 4)=6[/math]
Then for any [math]r\in[0,1][/math], all Cauchy sequences [math]q_n:=\frac{1}{a(r, n)}[/math] are in the Cauchy real [math]0[/math].

Now usually one would say that conversely, the cardinality of [math]{\mathbb Q}^{\mathbb N}[/math] is also dominated by that of [math][0,1]=2^{\mathbb N}[/math], but I'm actually unsure if in the right framework, it might not be possble for [math]{\mathbb Q}^{\mathbb N}[/math] to exceed the interval. Anybody got input? I understand that "right framework" is a bit vague.

>>12189241
If you write [math](a,a,..,a)[/math], this suggest more that you speak about a pair than a Cartesian product in the sense of [math]a\times a\times a\times \cdots \times a[/math]. What is it? What's your [math]a[/math]?

- The latter is actually in bijection with the maps of the n-element set into [math]a[/math], e.g.
[math]|a\times a\times a| = |a|^{\{0,1,2\}}=|\{0,1,2\}\to a|[/math], because each amount to choosing 3 elements of [math]a[/math].
This brings us back to coding questions btw., the recursive definition of tuples requires Replacement if you go too far. I think the function space definition, on the other hand, needs less

- The former (mapping some element [math]a[/math] into the pair holding three copies of [math]a[/math]'s), is the map into the "diagonal", or comultiplication. That's a name for something like that in Hopf algebras anyway.

[Anonymous No.12189273]

https://en.wikipedia.org/wiki/Hopf_algebra#Examples
Here's a few of those, not sure if this is what you mean, but the more you know...

[Anonymous No.12189276]

>>12183826
>additional requirements
I meant that for EU citizens if you've a graduation that qualifies you to study at university in your country, then that's enough. For math you probably don't need to bother with DoSV because at most universities math is admission-free ("zulassungsfrei"). Which uni do you want to go to?

[Anonymous No.12189279]

when I say pair I mean tuple

[Anonymous No.12189283]

>>12189258
yes I'm just not sure if it's stupid or if there's a better choice.

>>12189268
[math]a[/math] is an element of some set [math]A[/math] I want [math]a^{\times n}[/math] to be an element of [math]A \times \ldots \times A[/math].

[Anonymous No.12189286]

>>12189283
I would use it. It tells everything there is to say about your element. It just n copies of a like that.

[Anonymous No.12189288]

>>12189283
Then I'd define it at the start of the text as the anon said, and I'd call it \Delta, simply because I've seen that in some contexts, but whatev

[Anonymous No.12189413]

>>12183906
>Hot, feminine
>Highly intelligent
>Enough discipline and self-control to get a university education, in a male-dominated field, and stay attractive

>Wanting to cum on, instead of inside

Darwin Award nomination!

[Anonymous No.12189503]

>>12183092
openstax.org/subjects/math

[Anonymous No.12189507]

>>12189102
Video format is boring openstax.org is superior though.

[Anonymous No.12189628]

I want to beat up Nikolaj

[Anonymous No.12189630]

>>12189507
ty for the nice reference, maybe I will use it to learn physics. Khan does go pretty slowly and repeats himself a lot.

[Anonymous No.12189633]

>>12189628
i want nikolaj to feed me estrogens

[Anonymous No.12189689]

>>12189288
this

[gaobizi No.12189769]

>>12182912
>Graph theory is the slums of topology.

No, when Kuratowski proved the graph planarity theorem, depending on K5 or K3,3 minors, he said it showed there's more to life than topology. Graph theory isn't part of topology at all. It's purely combinatorics.

[Anonymous No.12189779]

>>12189769
>There's more to life than just topology, there's combinatorics
That's like saying Heaven isn't the only thing, Hell exists too!

[Anonymous No.12189787]

>>12187700
Sorry for getting back late anon, I had no internet access last night. I'm going to cover the following topics:
>fractions and base 10
>parabolas and distributive property
>exponentials
>sine, cosine, tangent
>differentiation and integration
So first up is fractions. I will explain what they mean and how to compute them. A fraction can be thought of as either division, or as a ratio. 2/4 as division breaks 2 into 4 parts, which means each part is size 1/2. Ratios give the same result, because the existence of 2:4 means for every one unit (2), there are 2 units (2*2=4), the base rate is 1:2. A fraction is equivalent to its other forms because the numerator and denominator are just scaled by the same factor. 1/2 is half the size numerator, broken into half the parts, compared to 2/4, so the end result is the same piece size. Now, here is how you can compute any fraction into a decimal. For example, we can use something complex, like 1/237. We can do something sly here, let's first find out how many times 237 goes into 1000. It goes in 4 times, because 250*4=1000, and each 237 is 13 less than 250, so after 237 goes in 4 times, there is 13*4 =52 pieces left over. This means 1000/237 = 4*237/237+ (52/237), which is just 4 + 52/237. This turns the improper fraction 1000/237 into a proper fraction, as well. However, we wanted 1/237, not 1000/237, our answer is 1000 times too big! So what do we do? Simply divide our answer by 1000, (4+52/237)/1000. Because we have two separate pieces, we can divide each by 1000 individually as 4/1000 and 52/237*1000, or sum them and divide after. It is easier in this case to sum then divide at the end, as you will see. The process we did for 1/237 can be done for 52/237, except instead of needing to multiply the numerator by 1000, we only need to multiply by 10 to make 520, the first multiple that's greater than 237. 520/237 = 2+46/237, all divided by 10 leaves us so far at 4+((2+46/237)/10)/1000.

[Anonymous No.12189792]

>>12189787
>contd

Carrying out this process looks like:
>1000/237 /1000 =
>4 + 52/237 /1000 =
>4 + (520/237 /10) /1000
>4 + ((2 + 46/237) /10) /1000
>4 + ((2 + (460/237 /10))/10) /1000
>4 + ((2 + (1+ 223/237)/10)/10)/1000...
The important thing to notice is the whole numbers and how many divisions of 10 they have under them. 4 has been divided by 1000. 2 has also been divided by 1000, but also by 10, and 1 has been divided by 1000 and 10 and 10. It looks messy here but when written out as a fraction where you simply keep each layer on top of the power of 10 that divides it, all you have to do is count the 0s of the 10s to find out how many 0s come before a given whole number in this expansion in a decimal (including the 0 before the ., as in 1/1000 is 0.001, three 0s). I would draw it but my pen ran out of ink. I will talk about parabolas, the distributive property, and functions in general in my next post.

[gaobizi No.12189795]

>>12182555
>[Map of Mathematics, with "Category Theory" as a moon of abstraction floating over all]

If "Allegory Theory" (Freyd & Scedrov 1990) is added to the map, should it be:

A. a higher moon or asteroid floating above Category Theory, being more abstract and general?
B. a part of the Category Theory moon, being merely one application of Categories? or
C. territory somewhere on the ground, being close to relational and cylindric algebras?

[Anonymous No.12189807]

>>12189276
I want to go to Bonn. On their website in english, it said it is not zulassungfrei, but that it has a local admissions restrictions - no mention of DoSV though. On the german version, it mentioned DoSV.

[Anonymous No.12189824]

>>12188667
Who said I am an atheist? How do you know what my criterion for agreement with nature is?

>>12187595
I'm not much of an analyst, but aren't there multiple smooth curves between any set of points

>>12188175
You sound like a little rat creature. Like you're not a real human, just a taunting psychopomp from an old fairy tale.

[Anonymous No.12189837]

>>12189769
>more to life than topology
That doesn't make any sense. Kuratowski's result can be stated as the failure of an embedding of certain type of space into the plane. That's topology.

[Anonymous No.12189843]

>>12189837
>i can restate a result using topological terms, so it's about topology
leave that mental gymnastic for your grant applications

[Anonymous No.12189846]

>>12189795
>cylindric algebras
Is this still studied?
I mean the same question goes for allegories, I guess.

[Anonymous No.12189857]

>>12189843
Okay, so what is combinatorial about the proof?

[Anonymous No.12189866]

embeddings of graphs is definitely topology, lol

[Anonymous No.12189880]

>>12189857
here's the first proof that showed up on google
http://math.uchicago.edu/~may/REU2017/REUPapers/Xu,Yifan.pdf
show me the part which is not combinatorial

[Anonymous No.12189930]

>>12189787
>>12189792
Thank you so much anon, this definitely is going to help.

[Anonymous No.12189932]

>>12187402
just remember [math]det(A)=\sum_{\tau \in S_n}sign(\tau)a_{1,\tau (1)}a_{2, \tau (2)}...a_{n, \tau (n)}[/math] so [math]a_{1, 1}a_{2, 2}-a_{1,2}a_{2,1}[/math]

[Anonymous No.12189939]

>>12189807
I see no reason why DoSV shouldn't apply to foreign students too. Bonn is a good choice for math btw

[gaobizi No.12189942]

>>12189837
>Kuratowski's result can be stated as the failure of an embedding

CAN be stated....but NEED NOT be so stated. His planarity testing required no embeddings, etc.

I guess graphs can also be "seen as" one-dimensional manifolds, if that's what you like, instead of as purely combinatorial.

[Anonymous No.12189980]

>>12189880
>nuh-uh, you can't prove me wrong

[Anonymous No.12189981]

>>12189942
namefag doesn't know what a manifold is :^)

[Anonymous No.12189983]

>>12189981
Shocking stuff. It's almost like people namefag just for attention even though they have nothing of value to add.

[Anonymous No.12190004]

>>12189980
i just showed you a proof which uses nothing except basic graph-theoretic notions
i can't understand why would someone insist it's not "combinatorial"

[Anonymous No.12190016]

>>12189843
>>i can restate a result using topological terms, so it's about topology
yes, that's exactly how it works. something can be about topology and combinatorics at the same time, you know.

[gaobizi No.12190035]

>>12189795
>If "Allegory Theory" (Freyd & Scedrov 1990) is added to the map, should it be:
>A. a higher moon or asteroid floating above Category Theory, being more abstract and general?
>B. a part of the Category Theory moon, being merely one application of Categories? or
>C. territory somewhere on the ground, being close to relational and cylindric algebras?

I like A, because relations are more basic and general than functions. CT for "the Working Mathematician" is based on functIONs, functORS etc. because functions are mostly what mathematicians use practically day to day. But in the big picture, that may be a bit "parochial".

The grand story seems to be in N-ary relations among systems of N-ary relations, (legacy of Peirce, Schroeder, Tarski, B. Russell et al.) rather than just functions on functions.

Also, adjointness of categories, like a Galois Connection between two posets, is a duality. Studies at Darmstadt and Prague have developed "triadic Galois connections" among three posets. Has the same been done for three categories/allegories?

[Anonymous No.12190041]

>>12190035
What you're doing right now should be a bannable offense.

[gaobizi No.12190057]

i want to suck a dick of an anime tranny

[Anonymous No.12190131]

>>12190057
Ask the useless phd anime.

[gaobizi No.12190220]

>>12190035
>Has the same been done for three categories/allegories?

I.e., maybe something like some sort of "tri-adjointness" among them?

[Anonymous No.12190284]

>>12190131
>useless

[Anonymous No.12190341]

>>12190284
I am.

[Anonymous No.12190358]

>>12190341
Cheers, lad.

[Anonymous No.12190395]

>>12189824
>I'm not much of an analyst, but aren't there multiple smooth curves between any set of points
yes, but after setting a few properties you want your curve to hold you often pigeonhole yourself surprisingly quickly down to just one possible curve
for example: pic related is the set of conditions that gets you the gamma function as the curve filling in the non-whole number parts of the factorial

[Anonymous No.12190399]

Consider a partially ordered vector space [math]V[/math]. Let [math]f: V \times \cdots \times V \to V[/math] be a multilinear mapping such that, if [math]a_i \geq 0 \forall i[/math], then [math]f(a_1, \ldots, a_n) \geq 0[/math]. Show that if [math]a_i \geq b_i \forall i[/math], then [math]f(a_1, \ldots, a_n) \geq f(b_1, \ldots, b_n)[/math].

Is there a simple way to show this? The only way I can show this is to write out an ungodly number of summations over all the different indices.

[Anonymous No.12190458]

>>12190358
Cheers, m8.

>>12190399
>Is there a simple way to show this? The only way I can show this is to write out an ungodly number of summations over all the different indices.
Not really, but the horrible sum is actually not that bad. Take f(a stuff) - f(b stuff) and write it as a sum. Whatever you have as the "smallest" parts will be non-negative, and so will be all their sums. The sums themselves aren't even that important. Only the compatibility with your ordering.

[Anonymous No.12190470]

>>12189942
>I guess graphs can also be "seen as" one-dimensional manifolds
utterly btfo'd by a single letter:
X

[Anonymous No.12190530]

>>12190470
kek

[Anonymous No.12190542]

>>12190399
did you forget about something? maybe all [math]b_i \geq 0 [/math]? because as you've written it, it's not true

[Anonymous No.12190550]

>>12190542
yes sorry it's [math]a_i \geq b_i \geq 0 \forall i[/math].

[Anonymous No.12190582]

>>12190470
CW-complexes, then

[Anonymous No.12190594]

>>12190550
here's a way you might like:
[math]f(a_1, \dots a_n) = f(a_1 - b_1, a_2, \dots a_n) + f(b_1, a_2 - b_2, a_3, \dots a_n) + f(b_1, b_2, a_3 - b_3, a_4, \dots a_n) + \dots + f(b_1, \dots b_{n-1}, a_n - b_n) + f(b_1, \dots b_n)[/math]
(start summing from the end to the back to see the equality)

but ultimately i agree with the previous anon, the huge sum is not that bad. if it's a test, i would just write something along the lines of "expand everything and each term of the ungodly sum is nonnegative", without writing out the multiple indices

[Anonymous No.12190640]

>>12190399
>>12190458
>>12190594
brainlet here, what horrible sum are you guys talking about?

[Anonymous No.12190670]

>>12190640
in [math]f(a_1, \dots a_n)[/math], write each [math]a_i[/math] as [math] (a_i - b_i) + b_i[/math], then expand everything, so you get 2^n terms

[Anonymous No.12190681]

>>12190640
The function in the anon's post is multilinear, which would mean for 3 variables that [math] f(a-x, b-y, c-z) = f(a, b-y, c-z) - f(x, b-y, c-z) = f(a, b, c-z) - f(a, y, c-z) - f(x, b, c-z) + f(a, y, c-z) = f(a, b, c) - f(a, b, z) - f(a, y, c) + f(a, y, z) -f(x, b, c) + f(x, b, z) + f(a, y, c) - f(a, y, z)[/math]. As you see, it gets pretty big, 2^n as >>12190670 mentioned. I know there is a mistake somewhere probably maybe I believe, but anyway.

[Anonymous No.12190750]

The definition of a living creature
Is the biological organism commonly reffered to as "the being" such as for a human named Me, that organism is my body
IN ADDITION TO
The related components required to keep that being able to survive and feel well for its natural lifespan
So for a human
That entails land
A human in a empty box is not a human at all. Because that human has no resources and will die in 2 days of dehydration.

[Anonymous No.12190846]

>>12189930
So, let's get on to parabolas. First we need two prior concepts, which are called "lemmas."

>lemma 1, functions
What is a function? A function is simply an input-output device, with a certain rule that each input can give at most one output, but a specific output could come from multiple different inputs. A function has a "domain", which you specify as the whole group of numbers that you input. For example, "x^2" for the domain of x=5 is just an input of a single number, 5 (and an output of 25). However, commonly, wide domains such as "all real numbers" or "all positive reals" or "all positive reals in addition to 0, aka all nonnegative reals" are used. This means that x^2 on this domain is a set containing each member of the domain, such as each real number, and each output. It is not just one x and one y, it is all variable x, and all the corresponding y. Now, when you typically see a graph of a function on a piece of paper, the horizontal axis is called the x axis, corresponding to inputted x. The vertical axis is the y axis, and this is the output - you take a horizontal coordinate, find the number for its output, and raise it to that height. However, there are other ways to graph functions that are not commonly seen in 2d, but are preferred in higher dimensions (such as mapping a 2d plane to a 2d plane) to make it easier to visualize. In our case of mapping one dimension of x to one dimension of y, we can think of having a line with evenly spaced numbers, and stretching it onto a new line to correspond. So for x^2, the number that starts at 5 would be stretched to the position where 25 was. But this is not common until higher math. Lastly you can think of a function purely numerically, no graph, just as the set of inputs matched with each one's own output

[Anonymous No.12190851]

>>12190846
>cont'd

>lemma 2, distributive property
This will be useful in factor polynomials as you might see in (x+1)(x+2) = x*x + 1x + 2x + 2*1. commonly referred to as foiling, and its inverse. The reason why foiling works is due to the distributive property. This simply says that for any random numbers, a b and c, if you have a*(b+c), then you have a*b + a*c. The reason why this is true is due to how scaling (multiplication is just scaling and in vector math, normal numbers are called "scalars" even, since they can only act on vectors as multipliers). Imagine you have a bunch of blocks of "unit size" - they are all equivalent and their volume is referred to as the base volume, the unit volume. Imagine you count out b blocks, and c blocks separately. Now imagine you somehow grow each block by a scale of a, each block is a times larger. Then the total volume would be a*b + a*c. Now put those blocks together, and grow the total area by a factor a. Since each block is size 1, multiplying each block to be size a will make the whole area a times larger. In the process of accomplishing a*(b+c), you have done the same as before, multiplied each block that was from b by a, and each block that was from c by a, so a*(b+c)=a*b + a*c.

When you do the distributive property on a polynomial, you are doing it twice. (x+1)(x+2) can treat one of the terms as a unique number on its own, so you get (x+1)x+ (x+1)2. Then you just distribute each part again, getting the full factorization. You might object, how can I distribute x if x is a variable that refers to a domain and not just one number? The reason is because that for any value that x might take, when it's at a certain number, the distribution works like normal. So if the distribution is correct for all x in the domain, it is a suitable replacement for the original statement.

Next up is the quadratic formula, inverse distributing/foiling, and some facts about parabolas

[Anonymous No.12190853]

How exactly are natural transformations defined when the foundation is ZFC?

There are two options and I've seen people stating (sometimes with implicit details) each of them. Let [math]F, G[/math] be functors from the category [math]{\cal{C}}[/math] to the category [math]\cal{D}[/math], both of these being classes (Jech's sense). Then a natural transformation [math]\alpha : F \to G[/math] from [math]F[/math] to [math]G[/math] is:

1. A class function [math]a\mapsto \alpha_a[/math] from [math]\cal{C}[/math] to [math]\bigcup\limits_{a\in{\cal{C}}} {\cal{D}}(F(a), G(a))[/math] such that [math]\alpha_a \in {\cal{D}}(F(a), G(a)), \ \forall a\in \text{Ob}({\cal{C}})[/math] and [math]\alpha_b \circ F(f) = G(f)\circ \alpha_a, \ \forall f\in{\cal{C}}(a, b)[/math];

2. A class [math]\alpha \subseteq \bigcup\limits_{a\in{\cal{C}}} {\cal{D}}(F(a), G(a))[/math] such that [math]\alpha \cap {\cal{D}}(F(a), G(a)), \ \forall a\in\text{Ob}(\cal{C})[/math];
for every [math]\left(a\overset{f}{\to}b\right)\in \text{Hom}(\cal{C})[/math] there are [math]g\in\alpha\cap {\cal{D}}(F(a), G(a)), h \in \alpha\cap {\cal{D}}(F(b), G(b))[/math] such that [math]h\circ F(f) = G(f)\circ g[/math];
for every [math]a\in\text{Ob}(\cal{C})[/math] and every [math]g\in \alpha\cap {\cal{D}}(F(a), G(a))[/math], there exists [math]\left(a\overset{f}{\to}b\right)\in\text{Hom}({\cal{C}})[/math] and [math]h\in \alpha\cap {\cal{D}}(F(b), G(b))[/math] such that [math]h\circ F(f) = G(f)\circ g[/math];
the analogous of the above line but switching [math]g[/math] and [math]h[/math] roles.


Both of these definitions have problems.
The first seens the most adequate since we work with them through associations [math]a\mapsto \alpha_a[/math], I'm not sure if this is really unnecessary, otherwise the second definition doesn't actually work on practice.
However, the problem with the first definition is that [math]\alpha[/math] won't be a set if [math]\cal{C}[/math] isn't.

[Anonymous No.12190865]

>>12190853
Expanding a little bit more on my last sentence:
Even if [math]\cal{C}[/math] is locally small, [alpha]\alpha[/math] won't be a set, will be a proper class. Therefore we can't form classes of natural transformations between functors from [math]\cal{C}[/math] to any other category. Not even a class with just one natural transformation. Can you see where I'm going with this? Notice that if you pretend it's a class and find a bijection from it to an actual set, it doesn't prove that it is a class nor a set. The proper class [math]V[/math] of all sets would provide a bijection [math]\{V\}\to \{\emptyset\}[/math], but that doesn't imply [math]\{V\}[/math] is a class nor a set.

[Anonymous No.12190875]

>positive integers

[Anonymous No.12190903]

>natural numbers

[Anonymous No.12190965]

>>12190851
To the anon I am explaining stuff to. I have a headache right now from the bus smelling like handsanitizer so I will post more later, it will be on the next thread

>>12182535
To the rest of you, I know it is early, but since I am inhaling nauseating and psychoactive fumes in this public transportation, I can't focus on 4chan, so I will just make the new thread now. Post in it when you please

>>12190956

[Anonymous No.12190998]

>>12189110
Yes, you would just get the same matrix, which is clearly not an inverse of itself

[Anonymous No.12191134]

>>12190399
>>12190458
>>12190542
>>12190681
>can't recognize an induction problem
the absolute state of /mg/

[Anonymous No.12191148]

>>12190998
for one, no, there are changes you must make to the matrix in addition to multiplying by the det. For two, there are many types of matrices which are self-inverses

[Anonymous No.12191388]

What's the best way to calculate the error between to vector? Like I tried the length like a dumb fuck just to rialise that the length of two unite vector is the same.

[Anonymous No.12191409]

>>12191388
I guess I should jsut take the cosine

[Anonymous No.12191677]

>>12191388
take the length of their difference

[Anonymous No.12191772]

>>12190853
>falling for the "everything is a set" spook
a natural transformation is just a certain collection of arrows that is natural

[Anonymous No.12191901]

>>12191772
It's not a matter of personal choice or belief.

>a natural transformation is just a certain collection of arrows that is natural

Nobody would discover the Russel paradox if they always thought that this kind of phrase is a definition. You need to be more specific with what foundation you're adopting.
Either way, I'm not asking about how it is defined.

Funnily enough, the same mathematicians who claim to be developing category theory over ZFC write the same as you did. If you say that it's a collection of maps, then there is no notion of components indexed by objects of the source category. It's like saying that a collection of real numbers is a real function.

[Anonymous No.12191907]

>>12190399
I think you need to assume something like [math]b_i\ge 0, \ \forall i[/math], take the case where [math]a_1 > b_1[/math] and [math]a_i = b_i, \ \forall i \ge 2[/math] for instance.

[Anonymous No.12192042]

>>12191901
just say collection of arrows one for every object in blablabla etc. If someone inquires, handwave some bullshit about proper classes and conglomerates till they stop caring. That's how everyone does it

[Anonymous No.12192089]

>>12190853
If you need to work with large categories, try moving to a conservative extension of zfc like nbg or feferman's zfc/s (see https://golem.ph.utexas.edu/category/2009/11/feferman_set_theory.html )

[Anonymous No.12193457]

any advice on studying for my GRE exam?
should I just spam flashcards for the vocab and practice simple calculus for the main exam then for the GRE math just go through their practice exam?
Is there more I can be doing?