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milnor sphere edition
>>11850693
previous
>>
Just an idea I had after doing my algebra exam.
Consider the set of rational numbers except 0, equipped with multiplication. Clearly this is a group.
Choose some prime p. The set of integer powers of p should be a normal subgroup.
Does this subgroup quotient Q in any significant way? What is the order and the index?
Does this quotient group have a name? I feel it could be connected to p adic numbers, but I haven't studied them.
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>>11855805
>Elliptic curves
>>
What do you think about this?
https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/
Also, is quantamagazine prestigious or pop tier?
>>
>>11855825
>Does this quotient group have a name?
Yes, Q.
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>>11855878
Shit problem.
Pop /sci/ tier.
>>
>>11855890
Based.
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>>11855878
I am not a big math guy but it sounds like they just used vectors and linked them together on a graph.
Like, if the distance between these two points is this vector, and the distance between these two points is this vector, so there is probably an infinite number of these, and they just found a way to isolate the ones that intersect with eachother halfway.
But I honestly dont know so absolute grain of salt take here.
>>
How does one prepare for an oral exam?
I have my first oral exam in grad school. Just wanted to ask if you guys had any tips for me?
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>>11856177
didn't you have oral exams previously?
idk just learn everything as usual
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>>11856177
>>11856183
Where do people have oral exams?
>>
>>11856195
i have pretty much oral exams exclusively
t. polack
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>>11856204
What are they like? I only had written exams.
>>
>>11855825
>Q without 0
that's a free abelian group on countable infinite generators. by forming the quotient you described, you are just leaving out one of the generators. so you again have a free abelian group on countable infinite generators
>>
I've started reading Spivak as I haven't done calculus in years and I am going back to school for my masters. This example has me confused.

There I get the example and reasoning pic related.
>Theorem 3:
>If f is continuous on [a, b], then there is some number y in [a, b] such that
f(y) >= f(x) for all x in [a, b].
There is no y in [0, 1] because x can be a number so y<x<1.


My question is, why is the function f(x) = 1/x for x = (0, infinity) regarded as continuous but pic related not?
What's stopping you saying for all y in (0, infinity) you can chose an x so that 0<x<y and f(y) does not bound f(x)?
>>
>>11856209
honestly, they are way better in every aspect
>more flexible
if you're stuck on a problem, the prof usually gives you a hint or goes on to the next problem.
if your answer was too short or incomplete, the prof will ask you to elaborate.
Nethertheless, the prof is usually satisfied with short answers as long as he has the impression that you know what you're talking about. In short, you're allowed to handwave more (in moderation of course)
>it's way shorter
>you get your grade immediatly
>it's not bad if you make a mistake, as long as you can correct it after the prof told you it
the only real disadvantage is that it's too much effort if the class has too many people
>>
>>11856273
sounds fun, although I'd brobably umm and ahh too much trying to fill the silence while thinking
it can take me a good few seconds to properly process a difficult question/recall definition/theorems, especially when its not written down in front of me
>>
>>11856209
usually the examiner asks you "there was a theorem about X in the course, tell me the exact statement and rough idea of the proof"
some will also ask you to solve simple problems on the spot
i find them more stressful than written exams, but that's probably because it's harder to bullshit on an oral exam
basically this >>11856273
>>
>>11856273
>>11856283
Those sound quite comfy actually. Thanks for the replies.
>>
>>11855825
Not really, it is isomorphic to the group you started with. Order and index are infinite
>>
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>>11856209
In my experience they usually give you an exercice from those big books full of exercices from oral exams like pic related and you have 1 hour to solve it. Whether the examinator will help you or not depends on his personnality, some are huge fags and will even try to fuck with you by saying stuff like "are you sure about this statement?" when the thing is true.
>>
>>11856195
In french classes prépa, they have weekly oral tests in math and physics. You get to a blackboard and someone asks you about some aspect of the previous weeks’ lectures (prove such or such proposition/define this or that). After that, they ask you problem after problem for about an hour. You are encouraged to explain your reasoning every so often.

I know that Italian unis only have oral exams. From what I heard, they are similar to this >>11856273
>>
>>11856313
>>11856321
I see. Sounds like a lot of work for the examinator. Merci.
>>
>>11856334
>Sounds like a lot of work for the examinator
It isn't really, they usually spend the hour doing nothing, they get to relieve their stress on the students and they're payed between 50 and 80€ an hour.
>>
>>11856334
Yes and no. Because you get students by groups of 3, you need to look up dozens of problems every week for interesting ones (because you do not want to come up short if you get the occasional group of excellent students), which gets old and managing the class can be a pain in the ass. Sometimes all three get ideas at the same time and you need to time your explanations.
But yeah, overall it is a lot of fun and very well paid, as anon >>11856354 said, so it is a great gig. Much better than being at TA at university in many respects (although, being a TA now, I like having a regular group of students that I coach rather than a bunch of small groups that I meet three times a year)
>>
guys...
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>>11856406
I belif in u
>>
>>11856223
I could probably help you if I knew what the fuck you were asking.
>>
>>11856313
Source for this?
>>
>>11856223
interval [a,b] is closed
interval (0, infinity) is open
theorem 3 is about closed intervals
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>>11856500
Source for what?
>>
>>11856223
1/x is continuous everywhere in its domain.
0 is not in its domain so its not technically discontinuous there, but it is continuous in any small distance away from 0.
it just so happens that the domain is not connected, while an interval like [a,b] IS connected
>>
>>11856223
>Theorem 3:
>If f is continuous on [a, b], then there is some number y in [a, b] such that
This isn't the definition of continuity this is just a property of it. I believe the definition is: for any x, lim a->x f(a) = f(x)
>>
>>11856500
«Oraux X-ENS, Algèbre 1» by Francinou and Gianella (this one is availabie on libgen)
>>
About to submit my first paper

It has a total of 9 authors and we are submitting to a computational journal
>>
>>11856617
Congratulations anon! Hope you're under 24 though.
>>
>>11856617
Good job!
>>
what does it mean for two elliptic curves to be isomorphic
>>
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>>11856650
https://www.lmfdb.org/knowledge/show/ec.isomorphism
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>>11856659
what's an isogeny
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>>11856663
>>
>>11856621
>>11856631

It was a code monkey project and I mainly did it because I was rejected from all internships and REUs
>>
>>11856835
It's still more than I ever did, so good job.
>>
>>11856663
for elliptic curves: non-zero group homomorphism that's given by rational maps. really just the natural notion of a morphism between elliptic curves.
>>
>>11856650
intuitively: they are the same curve written in different coordinate systems.
>>
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Is planar geometry /mg/ approved?
>>
My hope is to go from undergrad directly to PhD. The full year algebra and analysis courses at my uni (which you NEED for any chance at this) are restricted to honors students only.

I didn't even know we had courses restricted to honors students or an honors program at my school. Apparently, they expect you to apply to this as a freshman. So, it isn't enough just to be a math major to take algebra and analysis. You have to be a sUpEr sPeCiAl math major.

In august when the instructors are announced, I'm going to have to email and beg the profs to let me take their courses. I couldn't enroll before they filled up because I'm not in this meme undergrad honors program. If they say no, I have no idea what I will do.
>>
>>11856918
Probably your master's, like most people.
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>>11856914
Of course
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>>11856918
Are you at Rutgers?
>>
>>11856918
Get a master's first. It's just another year or 2, and you can finish your PhD quicker so you usually finish the same time or maybe a year later. Also you have a better chance to get into a top school.
>>
>>11856959

UMN
>>
>>11856984
Ann Arbor?
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>>11856990
did the aliens relocate ann arbor while I wasn't looking
last I checked it was in michigan
>>
>>11856984
We have the same system at Rutgers. See if you can find the director of the honors program and email him/her explaining your situation. I was a transfer student and was able to join the honors program so if you can explain that you are serious about grad school, then I don't see why they wouldn't let you in.
>>
europoor here
in USA, how much money do you earn when you're doing a PhD?
>>
>>11856948
>>11856961

It wouldn't save any time. I'd have to spend next year taking filler classes, then take algebra and analysis in grad school. As far as I'm concerned, it would be a waste of time not to take these next year.

Also, a masters costs money. A PhD is also a job that pays you enough to live. So, I would not care how long the PhD takes.

As for getting into an elite grad school, I just don't care about that very much
>>
>>11856990

the university of Michigan is the U of M
the university of Minnesota is also a U of M
I said UMN, not U of M
You went with "university of Michigan"

>>11856997

That is good to know anon thank you
>>
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>>11857005
Zero (0)
They do it for free.
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>>11857005
Generally speaking between $20-25k USD per year, for an average student at an average school
it tends to be a little higher at top schools, although I think this is in large part because top schools tend to be in more expensive cities (e.g. students at Columbia or UCLA will get fatass stipends, but they don't really live any better because the city is so absurdly expensive to live in)
>>
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>>11857035
>They do it for free.
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>>11857036
> $20-25k USD per year
Isn't that below the poverty line?
>>
>>11857077
Not even close, that's like the poverty standard for a family of 3 or 4
>>
>>11857035
i'd do it for free
in fact, i do it for free, yes
>>
Supergeometry is a meme, isn't it? I can't take this anal rodeo of definitions seriously.
>muh anticommuting coordinates
>>
>>11857036
Is that including TA duties or just the bare minimum?
>>
>>11857219
Yes.
>>
>>11857261
Including.
What it is without TA duties varies a lot from school to school. At some schools your funding IS your TA pay, and the bare minimum without it is $0.
I can only speak from personal experience here, but on the admission offers I received where it was explicitly mentioned how much of my total funding was TA pay (I'm judging from 3 examples here), it was always roughly half.
You can also make a few grand more from 2nd year on if you agree to step up to teaching a whole class instead of just TA'ing it.
>>
>>11857300
Why would anyone want to teach a class?
>>
>>11856914
How do I into it
>>
>>11857326
money
good to have some experience teaching before you get thrown into the deep end in your first professorship
speaking of, it's nice to have on your CV when you try to find a job

and as long as you don't get saddled with precalc for ultra dummies it's not really that painful an experience
>>
>>11857326
Having sex with students.
>>
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Is information theory a meme?
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>>11857358
hehe i have this book picked it up at a used book sale :). information theory is not a meme but it's also not math
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>>11857372
>Information theory, a mathematical representation of the conditions and parameters affecting the transmission and processing of information.
Sounds like math to me.
>>
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>>11857386
Here's some information for you: yore a moran
>>
Is there a standard notion of a generalized inverse for operators on Hilbert spaces?

If [math]A[/math] is a non-invertible operator and you saw the notation [math]A^{-1}[/math], would you assume that this inverse is defined on the range of [math]A[/math] or the range of [math]A^{1/2}[/math]?
>>
>>11857408
Moore-Penrose tbqh.
>>
>>11857408
([math]A[/math] being positive in this particular example)
>>
>>11857416
AFAIK Moore-Penrose is only well-defined when the operator has closed range
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>>11857422
Yes, and?
He's not even specifying what inverse it is, might as well assume it's the most likely option and he didn't specify that the range is closed.
>>
>>11856223
as others have said, the ability to choose such a number is a secondary property of the function being continuous.
It is a big result relating to the notion of continuous functions being bounded on any compact set. The compactness is really important here (set closed and bounded)
1/x is continuous on (0, inf) because from the definition of the continuity (lim of a function as x -> a is f(a)) and the fact that the interval is open (no matter how close to 0, we will find that that lim of 1/x as x-> a is 1/a, because 0 is not included in (0, inf)).
and for the pic related: you need to go back to the chapter on real numbers and least upper bounds if you find this confusing. f on [0,1] is x^2 and suddenly jumps to 0 at 1. so its not continuous at 1, limit at 1 is not f(1) because f(1) is 0, and the limit is 1. As for the second notion that there is no f(y) that bounds every f(x) on [0,1], it follows from the properties of real numbers that you can choose real number z as close as you like to 1, that have f(z)= z^2, always. If f(1) was 1, then it would be continuous on [0,1] because the limit would be f(1), but f is 0 on 1 so you cannot choose f(1) as stated. You cannot choose something smaller than f(1) that would bound every f(x) in the interval, because f is increasing as x is increasing and from the properties of real numbers it follows that as soon as you choose some number close to 1 but smaller, you could find another one closer to 1, so f(another number) would be bigger that the bound you have found.
>>
>>11857408
>If A is a non-invertible operator and you saw the notation [math]A^{-1}[/math]
I would assume the author made a mistake. Generalized inverses come up rarely, if you are using them you should definitely mention what does it mean exactly
>>
>>11857408
>would you assume that this inverse is defined on the range of A or the range of A^1/2?
if A = T*T, then does it matter? you can define an inverse on the range of T and this gives you an operator with potentially a larger domain than if you defined it only on the range of T*T, no?
>>
Are there any fields of applied maths where you don't use gay stuff like analysis, stats or probablities and use topology, geometry or advanced algebra? Particularly interested if the field isn't theoretical physics.
>>
>>11857571
that's literally theoretical physics though
>>
>>11857571
crypto
>>
>>11857573
It literally isn't.
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>>11856195
For your oral exam, you must suck a multidimensional cock , so must be in the chamber of time and raep.
>>
>>11857587
>topology, geometry or advanced algebra
plenty of this shit in quantum computing for example
>>
>>11857573
yeah but fuck that, you're not making any money with it
>>11857583
number theory is even gayer than stat
>>
>>11857594
>quantum computing
>no probabilities
heh
>>
>>11857597
>you're not making any money with it
Yes, and?
>>
>>11857606
If I can't be a math professor, I want to be filthy rich.
>>
>>11857571
No, there isn't.
Geometry is inherently physics-biased, for applications.
Topology's the same.
Advanced algebra only has applications in crypto.

Also, even if you somehow find something, you can't apply geometry without piles and piles of analysis.
>>
>>11857604
what the fuck do you even want then
>has to be applied to but no physics allowed
>has to make me a lot of money
>no analysis allowed
>no stats allowed
>algebra is allowed but it can't have any numbers in it but it still has to be applied
What DO you like? You want somebody to pay you $300k starting to compute homology groups?
>>
>>11857618
>You want somebody to pay you $300k starting to compute homology groups?
Yes. This job doesn't exist?
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>>11857626
It does, it's called "professor at Stanford"
>>
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>Cryptography
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>>11857628
Yeah but I said applied math you dumb nigger
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>>11857636
No, there are no applied math jobs where you get paid an assload of money to sit there doing useless pure mathematics all day. Why did you even have to ask this?
>>
>>11857636
applied math professor then, you wh*toid
>>
>>11857571
I know where this question is coming from. No such a field does not exist.
You need to at least become competent in either analysis or stochastics.
It's practically impossible to avoid either of those fields in any area that isn't pure algebra/geometry/topology.
>>
>>11857644
Why the FUCK not?
>>
>>11857644
>>11857649
well fuck you then
>>11857664
I'm not against analysis per se, I just don't want to do exclusively analysis or probabilities all day.
>>
>>11857571
robotics
error correcting codes
cryptography
>>
>>11857669
Because it's not profitable.
>>
>>11857674
>robotics
can you expand on that
>error correcting codes
yeah true
>cryptography
booooring
>>
>>11857664
There's nothing wrong with stochastics.
>>
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>>11857597
>number theory is even gayer than stat
>doesn't like analysis
>doesn't realize that a solid knowledge of analysis is very important for literally every program of geometry study
>thinks that "advanced algebra" is a term that means something
I don't think you like math dude. You might try "full-stack javascript slave" at a bug-chasing start up; I think that's what you're looking for!
>>
>>11857688
I really don't get how you can enjoy mathematics but not enjoy number theory. Number theory is literally "the entire rest of math, used to prove cool little factoids about integers". It's just everything else with sprinkles on it. If you don't like number theory you don't like anything.
>>
>>11857688
>mumin oberst
That is actually a hemulen.
>>
>>11857679
https://www.idiap.ch/workshop/magkdr/
https://www3.nd.edu/~cwample1/Preprints/NumAlgGeomAndAlgKinematics.pdf
>>
>>11857702
extremely based post
>>
>>11857688
I mean, I'm pretty good at analysis, real, complex, functional, etc... I just doesn't like it.
By "advanced algebra" I meant stuff like commutative algebra, representation theory, galois theory and everything that isn't just basic algebra like linear algebra, basic group/ring theory, etc...
>>
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>>11857679
>>cryptography
>booooring
So do you like anything?
>>
>>11857711
double dubs don't lie
>>
>>11857721
Geometry, topology, and advanced algebra. You can't read?
>>
just ignore this retard
>>
>>11857731
>You can't read?
Yes.
>>
>>11857702
Yep, very well said
>>11857708
How do you figure? It looks more like a moomin on account of the ears
>>11857714
you'd be hard-pressed to find anyone doing pure algebra research, even in academia
One thing you could do is "topological data analysis" but I don't think it would be easy to find a job which allows you to do that
>>
>>11857679
>>can you expand on that
haha servo go brrr
>>
Should I just give up about trying to make money and only focus on pursuing an academic career?
>>
>>11857748
Because I grew up surrounded by moomin stuff. The two species are easily described by having the hemulens be strict followers of rules while the moomins are pretty much the opposite. That is from the DDR adaptation of the story where the valley is flooded and then the characters are spread all around the world. There is a garden guarded by hemulens where nobody is allowed to do anything, and that is its guard.
>>
What should I read after this
>>
>>11857799
Lang
>>
>>11857799
fuchs fomenko
>>
>>11857702
What are the prereqs to reading some basic intro to number theory book?
On that note, what's would be a good such book?
>>
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>>11857799
read some wikipedia pages about some topics you find interesting and when you find something you're very interested in find figure out what you need to study it formally and read those. it's ok if you lose track eventually as long as you're having fun and learning new things
>>
>>11857819
Hardy & Wright is a nice book and not "modern", so you can understand it out of the box.
>>
>>11857819
>What are the prereqs to reading some basic intro to number theory book?
basic abstract algebra, maybe field/galois theory
>On that note, what's would be a good such book?
ireland/rosen
>>
>>11857819
imo as long as you're fine with the idea of a field you can read niven zuckerman. if some of the algebra doesn't make sense to you just look it up in a decent algebra book (if you're a total noob, i reco gallian)
>>
>>11857819
https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_2?dchild=1&keywords=Number+theory+dover&qid=1593634190&sr=8-2
>>
>>11857819
Plenty of number theory books start from nothing
at all and could be read even by high school students (this kind of book is probably what you'll bump into first if you search "number theory" on Amazon), but I don't think that's a good idea since using literally zero algebra requires you to replace many intuitive proofs that use super basic algebraic tools with stupid ad hoc tricks. Even the crappiest undergrad algebra course is enough preparation to do things respectably.
>>
>>11857854
>could be read even by high school students
Kek
>>
>>11857702
I don't care about number theory because I am an unbelievable geometrytard and because I've never actually studied diophantine geometry.
>>
>>11857819
what is "
modular arithmetic, gauss reciprocity theorem and existence of generators mod p^k, 2p^k, basic divisility reasonings, linear diophantines, gaussian integers, quadratic forms, quadratic diophantines with two variables,- all of this can be covered with just high school knowledge but you need to be smart

>>11857854
i agree that a lot of things can be stated in terms of algebra
but in my opinion "ad hoc tricks" are the essence of mathematics, the good "ad hoc tricks" become standard tools later on
he should study basic algebra if he wants to git gud in mathematics in general, but if he wants to just learn some number theory, then he doesn't need it
you can replace a 3-line proof of fermat little theorem with "order of element divides order of group QED" but is that more intuitive to someone who is just starting to learn maths? i don't think so
>>
>>11857892
fuck i meant
>what is "intro to number theory" to you?
>>
>>11857892
>the good "ad hoc tricks" become standard tools later on
This is a true statement, but that's exactly my argument. The tricks needed to arbitrarily replace the fact that your students don't know what Lagrange's theorem is and have never heard of a finite field is are not "good tricks". They're shit tricks because you're deliberately not doing things properly.
>>
>>11857899
Anyone two weeks into an introductory algebra course knows what Lagrange's theorem is, though.
>>
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Given me a simple to understand problem (the statement of it being understandable), which can only be well solved with algebraic geometry methods post 1950.

Let me phrase it different: Give me a problem that Poincare and Hilbert would have stated, but where they would be interested and surprised about the method of solution.
>>
>>11857899
let p be prime, a is not divisible by p
the residues a, 2a, ... (p-1)a are a permutation of 1, 2, ... (p-1)
multiplying all the shit mod p, this gives you
[math]a^{p-1} \cdot 1 \cdot 2 \dots \cdot (p-1) \equiv 1 \cdot 2 \dots \cdot (p-1) [/math]
finally factor out the stuff to get [math]a^{p-1} \equiv 1[/math]

i can call this "a shit trick" as well as "a clever reasoning with simple tools"

my point is, if you are interested in natural numbers, then enjoy yourself - there is a huge number of facts that a beginner can understand without learning any auxilliary theory
and if you don't enjoy the "intro" stuff, then why do you think you will enjoy the next thing up
i think the problem stems from the pressure on many people - they feel they need to only study the "useful" stuff and quickly get through the basics to start research
>>
>>11857960
https://www.math.stonybrook.edu/~azinger/research/enumgeom.pdf
Does that count? Because I know jack shit about algebraic geometry, that's the only problem I can name that sounds like what you want.
>>
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>>11857960
Poincare conjecture
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>>11857972
Post more anime girls reading math books.
>>
>>11857960
look at "hasse-weil bound" here
https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves
i don't know shit about algebraic geometry but i can comprehend the statement, so i think it's simple enough
>>
>>11857979
I don't have that many.
>>11857972
Found what's probably a better .pdf :
http://www.math.utah.edu/~yplee/teaching/gw/Koch.pdf
>>
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>>11857985
>connection between theoretical physics (string theory) [...]
>quantum something something
I'm impressed that despite these buzzwords, this is a very interesting result
>>
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>>11857973
>>
>>11855878
Quanta is pop, but definitely the best pop out there currently IMO

>>11856997
wow, didn't expect to see someone else from Rutgers here. ru rah rah anon!
>>
>>11857981
Hasse might be a bit too old to fit

https://youtu.be/2nEzfa43VF8
>>
>>11857960
Weil conjectures
>>
>Problem has [math]\rho,p,\nu,v[/math] in it
>>
/sci/ I need your help. I made a digraph of the permutations of 4 elements and I'm trying to re-organize it into a solid. It has 24 vertices, 48 edges, and 26 faces. One caveat is that all cycles have to have a minimum length of 4
>>
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>>11858038
This is what I have thus far, I just need to figure out an elegant way to connect the redundant bits
>>
>>11858036
Also [mat]w, \omega[/math]!
>>
>>11858036
Also [math]w, \omega[/math]!
>>
>>11857960
Well defining moduli spaces (geometric spaces that classify other algebro-geometric objects) such as Hilbert schemes is a problem that dates back to at least Riemann but that Grothendieck completely reframed and made rigorous using his functorial approach. He also allowed for many such examples to be constructed with his two-punch deformation theory + algebraization combo. Before that, it was not completely clear that such spaces could even be defined and how.
>>
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>MFW trying to learn about wreath products
Lads, I think I might be a brainlet.
>>
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hi /mg/, how skilled do you have to be to achieve something like this?
>>
>>11858141
>r*ddit
>Skilled
Pick one.
>>
>>11858141
Low dimensional topology is very tricky.
Especially these kinds of proofs that are so obvious and yet so hard to put in words.

Don't underestimate a field just because the problem is easy to grasp.
>>
>>11858169
I will underestimate a field if some one from r*ddit managed to prove something in it.
>>
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>>11858036
I don't write rho, I always write p instead
>>
>>11858185
Opposite for me. Writing [math]\rho[/math] is inherently satisfying.
>>
>>11858141
Cute theorem but not actually impressive.
>>
>>11858197
What's an ugly theorem?
>>
>>11858222
We call them lemmas
>>
>>11858222
The ones that ask for five hundred different conditions and then give you a formula which is basically a formula you already have but a bunch of terms vanish because of the conditions.
>>
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>When you do a Cayley table, by hand, for a group with order 100
>>
>[math]Let \ M \ be \ an \ invertible \ matrix. \ Show \ that \ G_M^* \ is \ closed \ under \ inversion.[/math]
Anyone have any idea what the asterisks refers to here?
>>
>>11858036
>>11858062
>>11858065
do you by chance have trouble telling different shades of colors apart and also do you have any difficulty with detecting differences in facial structure in peoples of other races you aren't usually exposed to? Asking for personal research I promise not to call you a brainlet.
>>
>>11858430
No idea what this G of yours is, but it most likely has some non-invertible elements and the asterisk means you remove those.

>>11858438
I'm the typo retard and I was just adding stuff to his list for him to hate.
>>
>>11858443
And partially because I find it irritating when people call little omega w. [math]^\frown \omega^\frown [/math]
>>
>>11858430
asterisk often means dualization/adjoint
not sure if that's the case here because there's really no context given but it's possible
>>
>>11858430
I was thinking "the multiplicative subgroup generated by the matrix", but then it becomes tautological.
Got context? Algebra I? Algebra II? Doing a PhD and found it on an article?
>>
>>11858482
It's algebra I and unfortunately, that's all there is to the question which is why I'm a bit confused. I wasn't sure if * had some sort of universal meaning that I wasn't aware of.
>>
>>11858498
Do you know what [math]G_M[/math] is, at least?
>>
>>11858498
I've seen it used for duality and multiplicative subgroups of, say, integers modulo n, or some other ring.
Wikipedia mentions it, for example: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n
In your case, it's probably that, and the question is literally tautological.
>>
>>11858501
Going through my notes on Lie groups, I have that [math]G_M(K)=\{A\in M(n,K):A^TMA=M\}[/math]. Where M is some fixed non-singular matrix.
>>
>>11858515
Bro, I'll be honest.
I have no fucking idea what does the asterisk mean in that case.
>>
>always type d regardless of context
>always handwrite ∂, even in prose
>haven't touched LaTeX in years
>>
>>11858574
[math]\partial[/math]
>>
>>11858558
Yeah, I'll just ask my professor tomorrow, since I'm just as confused. Thanks anyways.
>>
>>11858616
0 isn't even in the set dude
>>
>>11858621
bro I deleted that shit immediately soo dumb
>>
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>>11858623
Haha LOL! nice (lmao)
>>
Just remember if you can't speak english, french, german, japanese and russian you're ngmi
>>
>>11858767
What if I speak English and French?
>>
>>11858769
good progress, a great start, you'll just do alright. I bet you can do better though, why not try?
>>
>>11858767
I always wanted to learn Russian but I'm too lazy to put that much effort in. So much stuff is written only in Russian and unlike French or German it's pretty much totally inaccessible unless you actually know something of the language.
>>
>>11858773
Because the effort outweighs the benefits.
>>
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Does math have a starting point? An axiom for forming axioms?
>>
>>11858803
You should do it, it's totally worth it. The best feeling in the world is when a language goes from just noise and nonsense to something you understand, even just at a low level and you only catch some of the words.

>>11858806
Hard disagree
>>
>>11858811
I think there's something like this in homotopy type theory, but I don't actually know
>>
>>11858811
What about an axiom for forming axions for forming axioms?
>>
>>11858811
>>11858831
Asking the right questions.
>>11858820
I'm interested.
>>
>>11857571
Formal verification and functional programming for software development
>>
>>11858234
:(
>>
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>>11858767
i speak english german russian but i'm still ngmi because my grades are too bad to get into phd
>>11858811
modern mathematics is ultimately based on the categorical imperative
>>
>>11858924
>modern mathematics is ultimately based on the categorical imperative

You mean Kant's idea that we should behave as if our behaviour was to be the universal standard? Isn't that just a reformulation of the Golden Rule? How is that related to math?
>>
>>11858930
Oh no, someone in /mg/ knows philosobabble but not abstract nonsense.
>>
>>11858938
?
>>
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>>11858924
>>
>>11858930
Math is inherently philosophical.
>>
>>11858924
There you go see you're short two. You need a balance of expressive languages and autistic ones
>>
>>11858946
I agree, but the categorical imperative is a very strange starting point.
>>
>>11858943
¿Why are you even here?
>>
>>11858930
>Isn't that just a reformulation of the Golden Rule?
It's not
>>11858956
>¿
>>
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>The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
>>
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>>11858953
Kant -> Husserl -> Cantor -> Zermelo -> ZFC
origato
>>
>>11855805
i'm currently in an undergrad which has a regular amount of math in it (up to calc 2 mandatory, take a guess)

can someone explain what the 'point' of math is? i don't mean this is a denigrating way, just in a 'what conclusion am i currently working towards' kind of way

taking linear algebra rn, btw
>>
>>11859092
it's a global tabletop rpg campaign for autists
>>
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post comfy math videos
https://www.youtube.com/watch?v=88Q2v6FTSBI
>>
>>11858986
Man who never got out of his hometown sees no value in experience, more at 11
>>
>>11859103
lmfao
>>
>>11859114
Yes, and?
>>
>>11859109
https://www.youtube.com/watch?v=4t1mgEBx1nQ
>>
>>11859103
This.
>>
>>11859114
>t. Locke
>>
>>11859109
https://www.youtube.com/watch?v=zIEJSmcvs40&t
Ted Slaman lectures are pure comfy.
>>
>>11855805
I am *just* getting into formal logic.
I was wondering if the invalidity of regressive logic can be proven. (that, "P1 is true if is P2, which is true if P3 is... ad infinitum", "therefore P1 is true")
>>
>>11859219
sounds like proof by induction
>>
>>11859219
>categorical imperative
How on earth does an infinite chain of IF statements end with a THEREFORE?
>>
>>11859228
wtf? I must have had that text highlighted from earlier in the thread. Ignore the green.
>>
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>>11859219
most proof systems (all that i've come across) restrict proofs to a finite amount of steps
>>11859223
nope, proof by induction is a countable set of proofs with finite steps, he's talking about a proof with infinite steps
>>
bros...
>>
https://golem.ph.utexas.edu/category/2020/07/congratulations_john.html
>By that point, I’m not proud to say, I had flattened myself against the wall in a not-very-heroic manner. Our squirrels are small, but they’re wild animals and I guess can defend themselves if they want to.
Category theorists everyone...
>>
>>11859228
It's strange. My question is whether or not it is the kind of thing that can be proven false.
>>11859241
Ah, that's a shame. I was wondering if there's some sort of induction-like (as suggested by >>11859223) proof that can talk about infinity without taking an infinite number of steps.
>>
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>>11859260
the problem is that proofs need to end in tautologies, the operative word being end. also we can and almost always are "talking about infinity" when proving theorems, so this doesn't require infinitely many steps.
>>
>>11859260
>It's strange. My question is whether or not it is the kind of thing that can be proven false.
I'd say a statement that doesn't end isn't even asking a question in the first place.
>>
>read rudin
>try to read another text that is based on previous knowledge of analysis
>can't remember a single theorem
JUST FUCK MY SHIT UP
>>
Does where you go to undergrad matter for pure math?


It really feels like either option for a white
middle class male is fucked in this regard. Either get in enormous amounts of debt never paying it off and hope to make connections at Harvard or graduate with minimal debt but never come close to a Harvard undergrad education.
>>
I don't know if this is the right thread but what's an easy to pick up book about probability theory?
>>
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>tfw the most famous theorem in your name is referred to as a "trick"
>>
>>11859587
your value as a mathematician is determined mainly by your efforts and talent
if you can be the best student in an average university, then you're pretty gud
>>
>>11859587
Go to where it's cheaper for undergrad, provided it has minimal standards for quality. You don't want to study in a complete shithole. Big fish in small pond effect is real. You don't want to be below average among your peers. Being exceptional can highly motivate you and put you on great terms with professors/etc.
>>
Best book for pde and FEM?
>>
>>11859613
https://www.amazon.com/Probability-Theory-Concise-Course-Mathematics/dp/0486635449?ref_=ast_slp_dp
>>
>>11858767
What does ngmi mean?
>>
>>11859694
bro... you're not gonna make it...
>>
>>11859694
Ngmi (unironically).
>>
If my odds of winning £5 on a game that costs £1 to play are 1 : 6, is it correct to say that on average I will net £5 x 1/6 = £0.83 and therefore be ~£0.27 worse off in aggregate?

Similarly, if my odds of winning are now 1 : 0.6, is it correct to say that I will net, on average, £5 x 1/0.6 = £5 x 5/3 = £8.3 and therefore be ~£7.3 better off in aggregate?
>>
>>11859723
1 - 0.83 = 0.17, not 0.27
0.6 = 3/5, not 5/3
>>
>>11859587
wherever you go do not afraid of you're advisor
dont be that cunthole who wastes their office hours being stuck on an example set after only a couple of hours either. If your work is up to date you can pop in and say hi, remember this guy will be giving you a reference in a couple years

>>11859723
1. you win once and lose 6 times out of 7 altogether, try your calculations again
2. multiply both sides back up to whole numbers and try again
>>
>>11859725
>1 - 0.83 = 0.17, not 0.27
Yeah, my bad.
>0.6 = 3/5, not 5/3
I got 5/3 from 1/0.6, not just a 0.6 conversion.
>>
>>11859726
>try your calculations again
>and try again
Sorry man, don't know what I'm missing. What is wrong with this statement and conclusion?
>£1 game with 1:6 odds of winning £5 = £0.83 per play on average
>that is therefore a net deficit of £0.17 per game
>>
>>11859743
1 : 6 odds means "on average, you will lose 6x more often than win"
not "you will win one time out of six"
think about it, 1 : 1 means 50% chance
>>
>>11859626
not him, but i'm applying for undergrad maths at oxford. is it guaranteed i'll be the dumbest person there?
>>
>>11859747
Ah, okay.
So with 1 : 6 odds, I will win £5 x 1/7 = £0.71 per play on average (i.e. net loss of £0.29 per play)?

And, in the case where the odds are now 1 : 0.6, I will win £5 x 1(0.6 + 1) = £5 x 5/8 = £3.13 per play on average (i.e. net gain of £2.13 per play)?
>>
for a plane in E^3
say 4x - 5y + z = 20
why is t(4,-5,1) perpendicular?
>>
>>11857960
Unironically Fermat's last theorem. Hilbert even stated it as an important problem in his time
>>
>>11859795
What would you show them ironically?
>>
>>11859807
weil conjectures, poincare conjecture
>>
>>11859807
All episodes of The Melancholy of Haruhi Suzumiya in every possible order
>>
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>>11857972
i dont know how or why the author felt the need to define projective space right before this lemma
>>
>>11859753
>is it guaranteed i'll be the dumbest person there?
No, but it's likely that you will not stand out
>>
>>11859826
They were setting the notation for homogeneous coordinates, not defining the projective plane
>>
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Is there a classification of possible singularities on a plane curve (or in fact, any variety)? I can imagine it's completely resolved by just looking at the taylor expansion around the singularity, but i could be wrong. I'd also assume it exists, because of Hironaka's theorem, but i could be wrong
>>
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Anyone?
>>11859758
>>
>>11859853
Hi.
>>
>>11859758
>>11859853
yes you got it right now
>>
>>11859859
Great, thanks very much lads. Gonna gamble the last of my savings now.
>>
>>11859771
did you even bother checking the vector of direction my man
>>
How do I memorize theorems? How do you memorize them? Do you just learn the complete proof or merely its statement?
>>
>>11859984
so ax+by+cz=0
then I need two lines in the plane
choose ax+by=0, ax+cz=0
x=-by/a, x=-cz/a, then d/dy and d/dz respectively to find directions
(dx/dy=-b/a, 1, 0) and (dx/dz=-c/a, 0, 1)
then exterior product for perp. vector
(1, b/a, c/a)
multiply by scalar a -> (a, b, c)
I think this is right
neat
>>
>>11860059
carve them into your skin
the statement(sometimes name) are the minimum
the proof(s) can illuminate
an examiner might ask for one or both
>>
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>>11857972
>>
[math] \frac{d}{dx}x=1[/math]
>>
>>11860366
Uhh, hello, based department?
>>
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[math] \frac{{\mathrm d}}{{\mathrm d}x} (x\mapsto x) = 1 [/math]
>>
>>11860479
actually, I should even have written x mapsto 1
>>
>>11860227
Kek.
>>
any math lecture courses with a based lecturer like this?
https://www.youtube.com/watch?v=V49i_LM8B0E&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic
>>
Is there anything that's been written about the complexity of irreducibility testing in [math]\mathbb{Z}[x][/math] ?
It's fairly popular that we have "fast" primality tests for integers, but I haven't been able to find anyone talking about anything analogous for integer polynomials. Most of what I can find seems concerned with explicit factorization and/or lives in finite fields.
>>
>>11860479
lolis
>>
>>11860719
i'm sure a good google search will give you an answer
>>
I fuckin hate probability
>>
>>11860766
i hate how easy it is to misinterpret problems
>>
>>11860719
https://math.stackexchange.com/a/1978/248205
i think this gives you a practical algorithm which uses integer primality testing as a subroutine
but analysing the computational complexity is hopeless here
>>
>>11860777
Checked.
>>
>>11860690
>Quotes Wittgenstein less than a minute in
Holy based.
>>
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>>11860840
>>11860690
>>11860401
>>11857711
>>11855947
>>
>>11860849
As in (finitely) based, an equational theory generated by only a finite number of identities.
>>
>>11860902
Based.
>>
I am taking criticism.
>>
>>11860719
Ive seen it said that factoring polynomials over Z is much easier than factoring primes (I saw this in aluffis book, but no reference)
>>
Do all two digit numbers [math]10a+b;a,b<10;a,b\in \mathbb{N}[/math] possess the property [math]101a+2b=110a+11b[/math]?
>>
>>11860927
the Lax multi book sucks and you should swap out his Lin Alg book for Hoffman and Kunze. Shifrin or Hubbard and Hubbard if not Edwards or Lang for multi.
>>
>>11860934
what
>>
>>11860934
i think the only number that posseses that property is 0, so id say no
>>
>>11860927
Where the FUCK are elliptic curves, Lie theory, and representation theory?
>>
>>11860927
ignore what I said this chart is trash you are a retart for creating it
>>
>>11860927
>I am taking criticism.
It's shit, kill yourself.
>>
Who else is getting a PhD just to create the ultimate meme guide?
>>
>>11860943
Interesting, interesting, will flip through those.
>>11860947
>>11860948
>>11860957
Woah, woah, woah, calm down, I said I was taking criticism, not (You)s.
>>
if |A|=m then why does |{X⊆P(A):|X|≤1}|=m+1?
>>
>>11860971
empty set
>>
>>11860975
so there are m different sets X, plus the empty set?
>>
>>11860979
yes
>>
>>11860957
This.
>>
>>11860970
you should consider putting Munkres, Loomis and Sternberg or Folland for Advanced Calc/Multi desu
>>
>>11860971
as a side note, the reason its m+1 is because you wrote less than or equal to. There are m size 1 sets, and 1 size 0 set, the empty set
>>
>>11860944
It works with a=1, b=2, ie 12.
Split the 12 to 100 and 2, then add 1 and 2 and put it in the tens place.
100a+(a+b)+b=110a+11b.
>>
>>11860995
read back in the textbook, turns out i'd misread it and it was "X is an element of P(A)" rather than "X is a subset of P(A)" but i think the answer is the same
>>
>>11861001
what the fuck are you smoking
100*1+(1+2)+2 = 105
110*1+11*2 = 132

you're asking for 9a+9b = 0 but a and b positive
>>
>>11861001
Brainlet
>>
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>>11861017
>what the fuck are you smoking
>>
>>11860969
the average zoomer "scientist"
>>
>>11860934
by some arithmetic from your proposition follows that all 2 digit numbers z = 91*b, and b can be any digit 0 through 9, obviously only 0 and 1 are solutions so your proposition is false.
>>
>>11860943
Alright, Hoffman and Kunze and Lang made it in.
Anything else?
BTW, what is it that you dislike about Lax's multivariable calc?
>>
>>11860943
Hubbard was recommended to me and it seems nice.
>>
>>11861057
>what is it that you dislike about Lax
All the food stalls were super expensive (even for airport standards) and the washrooms weren't cleaned near enough.
>>
>>11861075
Underrated.
>>
the ol' reddit switcheroo
>>
a bit off topic, but has anyone here ever read Grothendieck's schizo ramblings?
>>
>>11861188
>schizo ramblings
Filtered.
>>
I'm proofreading grade 7 homework for my department and I've found something incoherent that has taken me over twenty minutes, that I still can't figure out.
>Here is a simple way to add hours and minutes together.
>As an example, lets add 1 hr and 30 minutes to 3 hr 55 minutes together.
>What you do is this:
>make the 1 hr 30 minutes into one number, which will give us 130 and do the same for the other number, 3 hours 50 minutes giving us 350
>Now you want to add these two numbers together
>130
>350
>----------
>480
>So we now have a sub total of 480.
>What you need to do to this and all sub totals is add the time constant of 40.
>No matter what the hours and minutes are, just add the 40 time constant to the sub total.
>480+40=520
>so we can now see our answer is 5hrs and 30 minutes!
I'm supposed to just be fixing typos and making sure that things are consistently abbreviated or not but this is fucking diabolical. Why are kids being taught this?
>>
>>11861233
Have you read it?
>>
>>11861259
>one hour plus one hour
>one hundred plus one hundred
>200
>plus forty
>240
>2:40
What the fuck is this?
>>
>>11861259
>time constant
>40
>>
>>11861277
Ah, this is just for when the sum of the minutes passes one hour.
Gee anon, you need to explain this properly.
>>
>>11861277
I don't know but this school board needs to fire the math teachers and start over.
>>
>>11861259
>>11861287
That's absolute fucking trash. Now wonder kids are still math illiterate in 2020.
>>
>>11861294
>Now wonder
>>
>>11861294
It is a fifth of a tenth of a millisecond faster than "just sum the hours and minutes separately and add everything together at the end" tho.
>>
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>>11861259
>the time constant of 40.
>>
>>11861259
I want to believe you just made that all up.
>>
>>11861277
>>11861287
>>11861294
The school department I'm at (basically a school board if you are confused by terminology; 6 schools share resources and management) can't be swayed by what I have to say, this is just my first year. I've seen the arguings by a guy who has been here for six years that schools need to stop using sets, numbers and integers interchangeably to no avail, like fuck will I get my foot in the door.
>>
>>11861338
Nice LARP.
>>
>>11861338
>like fuck will I get my foot in the door.
Extremely wise decision.
>>
>>11861330
bravo for creativity if he did
writing plausible-looking bullshit ain't easy
>>
>>11861306
Textual elision happens, get tover it.
>>
>>11861423
That why it probably is real - there are more retards on Earth than stars in the night sky.
>>
>>11861423
Yup. That is why I don't believe it is fake but sure do hope it is.
>>
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I was doing some exercises in Dummit and Foote and came across this one:
>If G is an abelian group and A is a normal subgroup, B any subgroup, and the intersection of A and B is a normal subgroup of AB.
Since AB is a subgroup of G, it is abelian, and because any subgroup of an abelian group is normal, the intersection of A and B is a normal subgroup of AB.

Is this all it is? I have the feeling that the proof shouldn't be this simple yet I can't see anything wrong with it.
>>
>>11861470
>and the intersection of A and B is a normal subgroup of AB.
THEN*
>>
>>11861470
With those assumptions it is correct. All you need to know is basically what you mentioned about the subgroups of abelian groups being all normal and that AB is a subgroup of G and hence abelian itself.
>>
>>11861470
Unironically, yes.
>>
>>11861470
I think the whole exercise is bust, its too obvious. all subgroups are normal, so the fact that he says a normal subgroup and on top of that another subgroup specifically not mentioning normal, makes me think its not what he intended. try removing the assumption that G is abelian
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>>11857799
oh hey I'm currently reading that.
I'm on chapter 4.4 and it's proceeding very slowly since I'm terrible at math.
But I'm doing my best to attempt all of the odd exercises and the interesting even ones and understand all the examples and solutions to the problems I don't get.
But it's still a slow process nonetheless.
I started reading it as a detour because I got to chapter 6 (groups of units) in pic related and decided that I wanted to brush up on my abstract algebra before proceeding.
Turns out I learned far less algebra in my classes than I previously thought.
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>>11861470
>>11861544
Bingo.
Quick solution: Take [math]p: G \rightarrow G/A[/math]. Then the restriction to [math]AB[/math] has kernel [math]A \cap B[/math], and we're done.
Filling in the gaps is left as an exercise to the reader.
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Is modal logic hard or am I just a fucking brainlet? I could mostly follow propositional modal logic but now I've been writing down (translating) the proof of validity of modal predicate logic in frames based on barcan formula models for like 7 pages without understanding a single fucking thing. I'm reading A New Introduction to Modal Logic from Hughes & Cresswell. Does anyone else have anything that might be a bit easier to digest?
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>>11861585
New thrrad
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>>11861579
Did you cover basic Lie theory and Burnside's lemma, at least?
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>>11857679
>cryptography
>boring
maximum cringe, but whatever

On that note though, I think pretty soon cryptography is going to sprout off from math and computer science and become its own field much in the same way computer science sprouted off of electrical engineering and mathematics.
A lot of recent developments seem to be towards developing non-encryption related uses like cryptographically verified voting systems, cryptographically verified push databases, zero knowledge proofs, cryptocurrency, etc...
Pretty soon there'll be a generalization of this "using crypto secrecy to implement trustworthy systems" concept and it will become the underpinnings of its own field.
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>>11861599
You're delusional, this isn't even CLOSE to being the case. Maybe in 50 years if things continue the way they are right now (they won't).
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>>11860366
ah yes.
[math] \frac{d}{dx}x = \frac{d}{dx}\frac{x}{1} [/math]
[math] \frac{d}{dx}\frac{x}{1}= \frac{dx}{dx} = 1 [/math]
Because the nominator and the denominator cancel each other out. Yeah, that is a classic.
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>>11861605
They will.
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>>11861610
[math]\frac{sinx}{x}=sin[/math]
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>>11861611
clearly not. Computing power is barely increasing each year, and the quantum meme isn't going to change a god damn thing about it. Plus that's assuming we don't undergo a MASSIVE anti-technology movement in the next year, which is extremely likely to happen.
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>>11861616
>616
>sin
numbers dont lie
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>>11861650
I'm an unironic LaVeyan Satanist.
AMA.
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>>11861677
what is your opinion on Deathspell Omega and theistic satanism?
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>>11861677
how numerous are academics in the organization?
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>>11861689
I'm not really familiar with them. I listened to Paracletus because I heard a lot of hype behind it, but I'm not much into black metal to begin with and it didn't really stick with me. That being said, Satanism, whether genuine or for just for the aesthetic, isn't uncommon within metal.

>>11861692
More than you'd think.
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>>11861677
Raised on any religion?
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>>11861727
I went to a Christian school, but my parents weren't religious at all and only sent me there because it was better than the alternative.
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>>11861586
That's a cool little solution. Thanks
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>>11861819
You're welcome.
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>>11861832
Thanks for replying to him for me.
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Is the scalar field associated with a vector space necessarily the same for a subspace?
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>>11857571
Computational Topology, like Persistent Homology.
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>>11862094
Yes
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Mathbros... I, a CSchad, kneel.



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