/sci/ - Science & Math

>>12097896
This will not be /mg/
Try again

https://www.mathtrainer.org/

Do math trainer and learn your multiplication tables, then learn division, then learn every operation with a base 10. Soon you will have superpowers and will be able to fly.

>>12097896
learn how to make the thread properly you fucking dope

>>12097896
pic made me laugh. lol

>>12097896
threadly reminder to ignore physicists

>>12097919
https://www.mathfluency.com/

Mufuggin multiplication tables n fractsheens

>>12097919
Based

>>12097908
Why not? Try what?

In Qing Liu's Algebraic Geometry and Arithmetic Curves, exercise 2.5.6 asks to prove that in a noetherian ring, if x is a regular element and [math]\mathfrak{p}[/math] is a prime ideal containing x that is minimal among those prime ideals that contain x, then [math]ht(\mathfrak{p}) = 1[/math].
Now, the height of p is either 0 or 1 because of the Principal Ideal theorem, so it's enough to prove that p is not a minimal prime ideal.
But that would mean that minimal prime ideals in noetherian rings contain only nilpotent elements, and that doesn't seem right, since the nilradical is the intersection of all prime ideals (and noetherian rings can have more than one minimal prime ideals if I'm not mistaken).

Does anyone see where I'm wrong?

There's nothing wrong with applied math.

>>12097896
Last thread was absolutely shambolic. Almost no discussion of actual mathematics at all.
Here's a problem for anyone who actually wants some maths posted:
Suppose [math]p(z)[/math] is an [math]n[/math]th degree polynomial with two of its zeros at [math]z=\pm1[/math] - show that [math]p'(z)[/math] must have a zero somewhere in the region [math]|z|\leqslant\cot\pi/n[/math]

>>12097945
Didn't link to old one and asked a question in op

>>12097945
what >>12098163 said, (previous >>12081118)
you also didn't explain what /mg/ stands for (should be included in your post, not just the
subject), and we have no idea which edition this is (because you didn't pick one).

Basically you broke every possible rule for starting a /mg/ thread

Can I get a recommendation for books or internet resources synthetic geometry beyond Euclid's Elements? I've already done most of them and it's very fun.

>>12098163
>>12098173
If you wanna have gay ass "rules" you can fuck off back to pleddit or tumblr or whatever shitty website of your choosing.

>>12098143
Hmm seems like something you solve using residue theorem, how far off am I?

>>12098211
Don't bring this newfag shit here. The OP was bad, so just shut the fuck up and lurk for a bit instead of trying to change 4chan to suit you.

>>12098218
That's one way you could attack it. You just have to come up with a way to get the correct bound (it can be shown the bound is sharp).

>>12098211
Shut up, reddit.

>>12098224
Dude the OP being bad has literally no effect other than making some of you cry about it.
It's correctly titled and that's pretty much all that matters. Chill out dude

>>12098234
lurk

>>12098227
I'm sorry, mi knowledge in complex analysis is limited by what I was taught in the complex calc class I took for my EE degree (currently working in my math degree).
So what does it mean for a bound to be sharp?

>>12098243
An inequality being sharp means it's not possible to get a better approximation.
E.g. here it means the theorem is not true if we had [math]|z|\leqslant\cot\pi/n-\varepsilon[/math] for any [math]\varepsilon>0[/math].

>>12098240
This.

>>12098248
Hmm thats pretty interesting.

Btw does Banach's contraction theorem generalize to complex functions? I suppose it does as the original formulation is done regarding general metric spaces right?

Anyone have any good recommendations for a books on group theory with some tricky/interesting problems?

>>12098234
Not having a Backlink is pretty annoying tbqh, on good mg's you can answer questions 5 days later

Let [math]S=\{\sqrt{p}: p \text{ is prime }\}[/math]. How do I show that [math]\mathbb{Q}(S)[/math]isn't a finite extension? I tried showing that the tower of extensions at each point has degree two but for this I need to show that [math]x^2-p[/math] is irreducible at the previous field and this is where I'm stuck

>>12098195
Geometry revisited
EGMO - Evan Chen, maybe
Not sure what your goals are

>>12098398
This is valid. Should've posted the backlink

>>12098464
How isn't the irreducibility trivial from the definition of a prime number? Am I just missing something here?

>>12098594
asterix, you people are very easy to get trap on math

>undergrad faggot retards are making /mg/
>can’t even link the old thread
L O L

>>12098600
Well do explain what the challenge is here.

>>12098464
it would suffice to give infinitely many distict automorphisms in the galois group over Q

I'm trying to prove the estimate [math]\left(\frac{\pi}{\epsilon}\right)^{N/2}e^{-{x}^2/4\epsilon}\leq\left(\frac{2\pi N}{\delta^2}\right)^{N/2}e^{-x^2/4}[/math] for [math]\epsilon ,\delta\in(0,1)[/math], [math]N\in\mathbb{N}[/math] and [math]|x|\geq\delta[/math]. I get that this is equivalent to proving [math]\frac{\epsilon 2N}{1-\epsilon}\ln\left(\frac{\delta^2}{2N\epsilon}\right)\leq \delta^2[/math] but then When trying to find a maximum of the expression in the left wrt to [math]\epsilon[/math] I get to an equation only solvable by the lambert W and everything becomes a bit of a mess. What I really want is to prove that the integral of the family of gaussians converge to 0 at any neighborhood [math]|x|\geq 0[/math] by using dominated convergence and the uniform bound I proposed I got by just fucking around lol. If anyone knows a better bound that would also work. Plz help this little things are stopping me from finishing my thesis.

>>12098703
at any neighborhood [math]|x|\geq\delta[/math]

>>12098606
Lateral thinking, many equations have the same results, find the result and change the equation.

>>12098703
What's your thesis on?

>>12098723
I'm just filling all the details in an article about a rigorous construction of the feynman path integral. This particular construction uses oscillatory integrals and these are defined as limits of integrals of Schwartz functions so I'm just trying to find the value of a particular one.

>>12098464
have you tried googling "square root of primes linearly independent over Q"

>>12098746
can you post the paper - sounds interesting

https://mally.stanford.edu/Papers/structuralism.pdf

>>12098746
can you post the paper - sounds interesting

>>12098211

>>12098947
https://arxiv.org/abs/0809.4112 It requires a lot of regularization but for QED there isn't much really.

>>12099047
Okay, thanks.
I donno, there's something about that guys writing style that I don't like

>>12099088

Is there a lot of grade inflation in grad school versus undergrad? My professor basically said "nobody will fail this class unless you do not submit problem sets / exams".

Is the assumption that the students are now all of sufficient quality to easily pass? Or is it just that otherwise poor students are inflated to a C?

>>12099114
Grad school is like whose line is it anyway. Everythings made up and the grades don't matter.

>>12099134
Did /mg/ ever find out a way to explain infentesimals to a 5th grader?

>>12099147
Yea, just tell them they don't exist

>>12099151
lol

>>12099112
erm okay

>>12099147
mg is not for kids tutors tho

>>12097910
Tried this and my wings already started to grow

>>12099114
If you get a C in grad school you're on thin fucking ice for getting kicked out. Hell, even in undergrad you basically only give a C when the student would've failed 50 years ago. If you fail a class now you don't have a pulse or literally never showed up.

Why is he such a procrastinator? Just like the average /sci/ poster.

>>12099155
Teaching is the most honorable profession

>>12099184
>American education
Tons and tons of people failed until kicked the fuck out at my uni, and this was in engineering lol

>>12099233

>>12099222
nice digits, I hate fags like spivak tho. They make math more accessible.

>>12099184
I thought anything under an A- in grad school was basically tantamount to failing.

>>12099231
okay, but still, I don't want to skim though trig on mg

>>12099252
what you be tellin me

>>12099294
Squares be magic

>>12099258
Not sure what you mean by that but it is a very strange textbook. He talks about trivial things and then gives complicated problems such as prove this and prove that which he doesn't explain how to approach and none of these problems are directly related to the material he briefly covered in the chapter. He is not the only one who does that. It is as if these kinds of books are to be used as a reference only when one studies with a tutor or something. They aren't any good for self study.

>>12099306
t. 120 iq that almost majored in math

In (b), what is meant by 'formula' and 'derivable'?

>>12098047
>But that would mean that minimal prime ideals in noetherian rings contain only nilpotent elements
Not really, it just means that minimal prime ideals contain only zero divisors

>>12099432
Why don't you share what you're reading for more context? What is Con here?

>>12099306
I thought Spivak was thought to be pretty good for self study though. It's somewhat gentle and has a solution manual

>>12099526
Conφ implies that a given set of formulas φ is consistent. A formula ψ is a 'certain string' over a set of symbols. This certain string is obtained by following certain rules. For example, given a symbol set {a, b, c, +, =}, you would want something like a + b = c to be a formula, rather than something like = a +.
If a set of formulas φ is consistent, then it's not inconsistent (lol) thus it is not true that both a formula ψ and its negation ~ψ can follow from φ. That's the gist of it, anyway. There's a lot more to it than that. If you want to know more, the book is called "Mathematical Logic" by Ebbinghaus et al.

The problem I am running into is how can there be a formula that is not derivable when formulas are formulated by derivation? Maybe there is something I am misunderstanding

>>12099641
>how can there be a formula that is not derivable when formulas are formulated by derivation?

I see. Formulas are jusy strings which follow certain rules. There could be a formula which does not follow from a specified set of "axioms", even though it is a valid formula.

>>12099718
To give an example, you might take the theory of semigroups, which has an axiom (xy)z≈x(yz). The formula xy≈yx is a valid formula, unlike something meaningless like ≈≈xy(≈(z, but you can't derive it from the associative law.

>>12099726
That really throws a wrench in things. How do you know xy≈yx is valid then? just cuz?

>The Summer of Corona killed /mg/
feels bad man

>>12099754
xy=yx is not valid in the theory of groups. The other anon is using 'valid' wrong. xy=yx is a well formed formula not valid, the group of real 2x2 matrices under matrix multiplication is an example of a group for which xy=yx is not true. Valid in logic means true in all models.
>>12099432
Godels second incompleteness says that the sentence con(PA) is independent of PA, thus if PA proved (or derived) con(PA), that is there is a finite sequence of forumulas with con(PA) being the last and each sucessive one is arrived at using a rule of inference, then PA would be inconsistent. A theory is inconsistent if it proves every sentence, thus this corrollary says that PA is consistent iff PA does not prove con(pA).

>>12099780
Oh, I see. Thanks a lot! That clarifies things.

>>12099780
>The other anon is using 'valid' wrong.
Whoops, I'm a budding algebraist so the logic terminology is still new to me, although I obviously understood what was meant here. Glad to see someone with a logic background showed up to help out! :)

How do you define area?

>>12099866
https://en.wikipedia.org/wiki/Measure_(mathematics)

>>12099872
What are the minimum requirements to get into measure theory?

>>12099872
Warning: In order to use this concept for the "usual" version of area you need to already know the formula for the area of a rectangle, so it's not entirely fundamental in the sense that it would explain why that formula holds.

>>12099879
Undergraduate analysis, mathematical maturity. Topology might help.

>>12099879
Basic analysis, depends on the book.
>>12099883
You need to at least take as an axiom that the unit square has area 1.

>>12097896
Wow, small penis energy that they break up (baby) Rudin into three courses. Jesus, Gen-Z has it so easy.

>>12097896

>>12099925
zoomers won't even have to take exams or qualifiers in the future. clown world

>>12099930
Based Uncle Ted.

>>12098966
t.

>>12100027
>t. newfag probably from reddit

>>12099925
omg dude you had it so hard, youre so incredibly smart and talented i cant even!

>>12098713
shut up faggot

>>12099506
Ah shit, you're right. Somehow I got confused with reduced rings where 0 is the only nilpotent element and thought a regular element was just a non nilpotent element.
Cheers!

>>12098143
Interesting bound. I'm thinking this has something to do with [math]\cot\pi/n[/math] being the radius of the incircle of a regular [math]n[/math]-gon, but I could be way way off here.

>>12100049
He's right, mathematics courses are lowering their standards hard and it looks a lot like what's in the OP

Can someone help me understand why Presburger arithmetic can't include multiplication for it to be decidable? Is that a general rule, that multiplication makes your axioms undecidable?

Hey you stupid fucking mathematician cunts would it fucking kill you to write out a goddamn sentence, with actual fucking words, once in a while explaining what the fuck it even is that you just fucking did or what the fuck I'm supposed to actually do with your stupid fucking abstract mathematical objects? I know you have some sick fetish/obsession with trying to explain everything with just numbers and symbols and using the least amount of words possible because you were probably such a fucking socially awkward weirdo your whole life that now you need to try and prove you're better than anyone at all making your fucking ideas unnecessarily obtuse and inaccessible to anyone but your snide little mathematician friends, but the only reason we keep you on the payroll is to come up with tools for us to use to solve real problems. Sure we let you dick around with your insanely nonsensical 97 dimensional semicuspidal manifolds or your hyperbolic triangles without embedded eigenvalues or whatever the fuck it is you assholes are doing all day. But that's just to keep you little shits busy so you don't bother the rest of us with your fucking bullshit periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves. Jesus Fucking Christ. All I'm asking for is ONE (maybe 2) sentences explaining what the fuck this shit even represents. You can blame it on me. Tell your math buddies I snuck it and wrote 'words' while you weren't looking. Just please, for the love of god, tell the rest of us what you are doing and what you want us to do.

>>12101357
>I know you have some sick fetish/obsession with trying to explain everything with just numbers and symbols and using the least amount of words possible
underage b&

>>12101359
>replying to pasta

What is the largest pair of prime powers that differ by [math]1[/math] known? (For example [math]2^{16}=65536[/math] and [math]65537[/math])

>>12101366
I guess it smells like pasta but I didn't know for sure. I'm here all the fucking time but I still can't keep up with it all.

Asked this question in the last thread but forgot: how do I keep my math skills after learning some new concept? It takes me about a week until I start to forget stuff. Studying trig at the moment right now.

is mathematical maturity a meme?
if not, how do you achieve it?

>>12101526
watch conferences about the topic you learnt

My uni prof made the most unwatchable lecture videos for differential equations course. The MIT OCW course is older than age of preference for a lot of politicians (14 i think), and I don't know how to read.

Can someone please suggest a learning resource for learning differential equations? Preferable videos to watch while high. Pls help

>>12101605
What do you think it's outdated information after 14 years? Just use the fucking MIT course.

>>12101609
But the camera quality is really bad...........

>>12101192
I think it will hinge on whether you can set up the Gödel [math]\beta[/math]-function and related tools, which tie to e.g. the Chinese remainder theorem and modular arithmetic.
These context's allow you to capture formulas with arithmetic. This goes in that direction:
https://youtu.be/soHBNEJlzL0

>>12101192
For your first question elimination of quantifiers is the proof technique used to show that Presburger is decidable, Enderton's Logic book section 3.2 or 3.3 has a very nice proof. For your second question, multiplication allows you to do coding. Multiplication allows PA (or Robinson's Q for a stronger statement) to reason about its own proofs. In particular there is a definable ternary relation in PA, say R(a,b,c), where PA proves R(a,b,c) iff b is the code for a forumla with one free variable, with a is some number and c codes proof that formula with input a. One consequence of this is that PA can reason about its own consistency, there is a universally quantified statement which says that for all n, n is not the code of a proof of 0=1.

I'm gonna read the EGA, wish me luck lads.

>>12101713
Why don't you go for a more digestible and synthetic source?
It would probably be more efficient.

>>12101522
2^82589933-1 and 2^82589933

How to get Fortran compiller on Windows 10??

>>12101586
>is basic competence with mathematics more modern than the 17th century a meme

Any of you's do any sort of tutoring?

https://arxiv.org/pdf/2009.03393.pdf

It's over for mathematicians.

>>12101774
because Grothendieck is based, it's one of the most complete ones, and I have enough time to not care about efficiency.

>>12102072
>Examples of equalities produced by the generator:
>(AA)[math]^2[/math]=A[math]^2[/math]AA
OH NONONO MATH BROS WE GOT TOO COCKY

>>12101790
Why didn't I thing about Mersenne primes? Thanks.

>>12101713
Good luck anon, but honestly you’re probably not gonna make it and probably will waste time that could be better allocated

Hey all, first year bachelor (well, soon to be second) math here. In my first semester I started out pretty well, but in the second semester the rona happened and I ran into a few difficulties. So now I'm looking for some advice regarding pure mathematics, more specifically analysis. I failed pretty spectacularly, and because it's a required course I'll have to retake it. Could any of you direct me towards a few books that might help me? It doesn't have to be about analysis specifically, as anything pure math could come in handy really.
The main difficulty I have is that I can understand the proofs and such that are in the book, but I have great difficulty when starting on exercises, maybe it just hasn't "clicked" yet, but in a lot of exercises they just seem to pull some magic example or such out of their ass, something I don't yet have the knowledge to do.
Also sorry for the blog post.

>>12102157
Velleman - How to Prove It
Tao - Analysis I
Abbott - Understanding Analysis
Gelbaum/Olmsted - Counterexamples in Analysis

>>12102116
But it also generates this!
[eqn]((A+B)^2)^2(A+A) = ((A+B)^2(AB+AB+AA+BB) + (A+B)^2(AB+AB+AA+BB))A[/eqn]
Any human would obviously get bored before expanding that far, and that's how the machines win.

Having trouble formatting/calculating f(4)-f(0)/4-0 into x^2 + 12x

Is the answer 4 or 16?

>>12102175
16

>>12099114
At least in my program anything < a B is considered failing.

>>12102180
thanks anon

>>12102172
You haven't done algebra until you derive a nontrivial identity that fills an entire page. I'm not joking.

>>12102170
Thank you very much, I'll check them out.

>signed up for business it because I suck at math and didn't do anything in 6 years
>first few classes of applied math
>I don't understand anything
How do I git gud fast? Real uni starts in about a week and they didn't tell me which things I am supposed to understand until three days ago. I haven't heard of about 60 percent of the things we do despite having done my a levels with fairly good grades. I'm kinda at a loss. Also the Prof doesn't really write down the formulas we use and instead immediately inserts numbers which makes learning and deeper understanding the formulas extremely excruciating.

Generally speaking, is linear algebra about active or passive transformations? What are we transforming, the coordinate system or the vectors in that system? My guess it is always the former.

(Apparently it is called alibi vs alias: never seen these terms ever used in any text books).

>>12102246
>alibi vs alias
I remember seeing those before.

But I don't think this is a question worth asking. It's about both? Wherever either of those notions (active or passive linear transformations) pop up, you're doing linear algebra.
If you're just asking what the standard array of examples are, then I suppose you'll find it out reading the texts.

>>12102254
I am just trying to reconcile the theory with a real world. Or a simulated world. For example a 2D video game. You have a 2D coordinate system which is the "world". And you have objects in that world. You can either transform/rotate those objects individually so their positions change relative to other objects. Or you can transform the entire coordinate system so all the objects will rotate but will remain unchanged relative to each other. So I wonder how this maps to the theory of vector spaces. It seems to that transforming the world maps well to the concept of the basis transformation. But when you rotate an individual object, should you also think that you are changing its basis in which case you'd have to treat that object as part of a separate vector space? In other words, should each individually movable object be considered a separate vector space with a separate basis in which case you would still be transforming the basis and the object would remain unchanged. My brain hurts.

>>12098143
I think you'll find [math]p(z)[/math] has to have real or imaginary coefficients for this to work.
>>12098218
>>12101044
Yes.

Does anyone know anything about the differential equations
[math] f'' f = -f [/math] and [math] f'' f^2 = -f [/math]?
The motivation is to imagine a car on [math] \mathbb{R} [/math] travelling towards 0 from the positive side with a starting velocity. it then brakes in a way proportional with the remaining velocity and inversely proportional with the distance. Does it stop before 0, at 0 or crash through it?
I included a case with f (the distance) being squared bc then the behavior on the negative side is the same.

>>12102392
damnit, the equations should be
[math] f'' f = -f' \quad f'' f^2 = -f' [/math]

>>12102297
They're 2 different but related things.
Changing a particular object within a coordinate system is basically multiplying by a matrix (applying a linear transform)
Changing the whole coordinate system is using different basis vectors (change of basis, and for each basis there's a change of basis matrix)

>>12102392
>>12102395

Consider the fact that

[math] (f \cdot f')' = 2f' + f\cdot f'' [/math]

and let me know if you can take it from here

>>12102421
mistake: there should be [math] (f')^2 [/math] instead of [math] 2f' [/math]

>>12102246
Lin Alg has nothing to do with coordinated. Your question is retarded, read a real textbook if you can’t see why.
>>12102254
Shut the fuck up faggot

>>12102395
Yeah, actually after I found my mistake >>12102430 I found there is no easy way to solve this and while it the solution is closed form it's a shitshow

[math] y(x) \sim W(x) = \frac{1}{\pi}\text{Re}\int_0^\pi \ln \big( \frac{e^{e^{it}} - xe^{-it}}{e^{e^{it}} - xe^{it}}\big) \mathbf{d}t [/math]

>>12102392
A more precise formulation of the problem: Let [math] f \in C^2([0,\infty) [/math] solve
[math] f(0) = p_0 > 0, \qquad f'(0) = v_0 < 0 [/math] and
[math] f'' f = -f' [/math].
Will it be the case that [math] f'(t) = 0, \quad f(t) = const [/math] after some point [math] t \geq t_0 [/math], or does [math] f [/math] somehow reach zero and a derivative or something becomes singular.

>>12102489
how do you solve something like this dude. I never had to deal with nonlinear ODEs before.

>>12102501
Yes

proof: look at the image

>>12102515
whoa

>>12102246
neither. Generally speaking, linear algebra is about vector spaces and linear maps between them (and systems of linear equations/matrices, but these can be interpreted as linear maps). The terms active and passive transformations only really make sense in for example in coordinate geometry but not in linear algebra per se

How would one go about this problem?

A function [math]f(x)[/math] defined for [math]x > a[/math] where [math]a[/math] is a constant, and [math]g(x)[/math] is a quartic whose leading coefficient is [math]-1[/math] that satisfies the following three conditions:

1) [math]\forall x \in \mathbb{R}[/math] such that [math]x > a, (x - a)f(x) = g(x)[/math].

2) For [math]\alpha, \beta \in \mathbb{R}[/math] such that [math]\alpha \neq \beta, f(x)[/math] obtains the same local minimum [math]M[/math] at [math]x = \alpha[/math] and [math]x = \beta. (M > 0)[/math]

3) [math]f(x)[/math] has more local extrema than [math]g(x)[/math].

[math]\beta - \alpha = 6 \sqrt{3}[/math]. Find the minimum of [math]M[/math].

Let's pretend that I am completely retarded with math and have to teach myself from scratch.
I have the following thing
(−17) · (−13) − |(−66) : (−11) + |−64| : 8| − |(−81) : (−27) − (−21)| =
The thing I wonder is whether || works similar to parantheses and the first and last, second and second to last and middle ones are part of the same unit or whether it's just 3 blocks of numbers inside of that.

>>12102662
Vertical bars usually denote absolute value.

Does anybody know what the function around t_j in the Jacobian here is? For reference, t_j is just a standard 3D vector. This is the first and only time this notation is used in the paper (ref: https://arxiv.org/pdf/1809.00952.pdf)) so I'm quite sure it's not something the author has come up with on the fly.

>>12102662
look carefully at where the + and - signs are, this will help you determine how to group the terms

>>12102172
is this true when entries of A and B are not commutative? or does it assume entries from R, C or some field K

You are a Italian mathematics student. You go to the first day of your Analysis course and this is the teacher.
What do you do?

>>12103415
What I always do in analysis classes: Take [math]\epsilon > 0[/math].

>>12102861
If you mean the brackets, it's mapping the 3D vector [math] t [/math] into the the standard so(3) representation, i.e.
[math] [t]_\times d = t\times d [/math]
https://youtu.be/gBMdTSXhYsY

I've come in contact with PGO and there's a fantastic 150 page writeup on the rotation involved, including all the Jacobians and they exotic addition and subtraction operations in tangent space that I glimpse in your paper. I can dig it out tomorrow if you will

What would you call an association that is like a function, but allows for one element to be mapped to several elements of its range? A map? An association?

>>12103703
They're called relations in English.

>>12103703
Relation

>>12103742
>>12103744
I thought relations were subsets of Cartesian products. This is not quite what I'm looking for. I will illustrate with an example:
Set A: {a,b,c}
Set B: {1,2,3}

A relation (over A x B x B) could then be (a, 1, 1) essentially mapping a to 1 and 1. However, I want to allow the following mappings equally:
a -> 1
b -> 1, 2
c -> 1
i.e. different sized tuples.

>>12103754
Seems to me that you want to map A to a tree on B. This is still a relation in the set theoretical sense. If for instance you mapped A into the tree of all finite sequences of B, then all elements would look like (a, (b_1,...,b_n)). That is just a functional relation on AxB^{<\omega}. You could replace omega with any finite or infinite ordinal really.

>>12099274
>>12099184
>>12102184
I guess what I mean by inflation is something like "students getting a better grade than they deserve".

I've never gotten less than an A in an undergrad math class and I'm still definitely not smart enough to be a mathematician. If the students in grad school are of a much better quality (and I'm at a school with a top 10 math program in the US so they definitely are) then sure, even without inflation, it would make sense that they all get As and Bs.

The alternative is that grades don't mean anything so A just means you had baseline competence.

>>12104089
This is for the US, right? At my uni we don't have grades, just fail/pass, and we still have a ton of people failing. I'd say a course has on average a 70% pass rate (Although it varies quite a lot between 40%-90%), and people who fail tend to just drop out.

>>12104103
What's the cutoff score for passing?

>>12104103
Yeah, I'm an undergrad at a US school. People definitely fail the undergrad level courses all the time. My understanding though is that due to the qualifying exams in grad school there's less need to actually fail students out of the program beforehand.

>>12104111
Depends on the course. For most exams it's 50% or more for a pass. This might sound low, but the structure is such that you're not guaranteed anything above that by just the material in the course. Basically the first half is stuff you can solve if you paid attention, and the second half requires you to be clever. Thus you need to remember the material perfectly if you're a smoothbrain.

In addition almost all courses also require you to do an oral examination as well. You need to pass both things to pass the course.

In contrast assignments and attendance account for 0% of your grade. You can literally never show up and never hand anything in, and still pass. Just show up and do the written and oral exams.

Also I believe our exams are longer than US ones? We usually have 6-7 hour exams, and I've heard, but I might be wrong, that US exams are around 90 min - 3 hours.

>>12104122
Normal exams are usually 75-90 minutes and finals are usually 3 hours. We also have no oral examinations until grad school and even then, I don't think they're everywhere. From my understanding, that seems to be a European thing.

>>12104122
Interesting. You mind sharing a bit more details about country / program? I've almost never had a math course in the US where this much of the exam could be completed via memorization.

>>12104137
Math program in Europe. Not sure what you mean by "memorization", mind explaining?

>>12104130
Yeah, it's really common over here in Europe. Normally oral examinations ask you to explain concepts, state and prove theorems that are part of the course, etc. Sometimes you'll be asked to prove new theorems, but often this is just a part of the written exam.

>>12104158
Where in Europe are you from? While anecdotal, both universities I've been at here in America have had a pretty noticeable portion of the math faculty from Eastern Europe, so I was wondering if there was a big emphasis on mathematics in those countries. A lot of them are from Poland, in particular.

>>12104158
>Math program in Europe. Not sure what you mean by "memorization", mind explaining?
Being asked to reproduce something that you saw in class or read in the book. If you had your notes and the textbook in front of you, this sort of problem would be trivial. Thus memorization of notes/textbook is sufficient for solving this problem. Obviously not necessary though as better students can fill in the details if they forget something.

Conversely, most exams I've sat have been largely structured so that even having remembered the material perfectly, or with the textbook/notes on hand, one wouldn't be guaranteed to get more than maybe 10% on the exam. The other 90% will involve varying degrees of 'clever' thinking.

What the fuck is the difference between GTM and Universitext?

Also does anyone have a recommendation for a concise review of ODEs?

I flipped a coin for 100 times and get 41 heads/59 tails. What is a probability that coin is crooked?

>>12103439
That would be very useful. The calculation of these complex Jacobians is by far the most annoying thing. It's tempting to do what solutions like g2o are doing and to just approximate using central difference, but the increase in accuracy from using analytic versions is too much to ignore...

>>12104418
100%

There is no such fair coin in the material world

>>12104462
lol

>>12104459
there you go

https://arxiv.org/pdf/1711.02508.pdf

>>12098047

take R\p it is a multiplicative set disjoint from (0) so...

I want to major in math, but don't want to work in academia. Is that possible?

>>12104515
yes after majoring in math, you can go to work for amazon

>>12104507
Thanks a lot. This is great as a reference and the IMU chapters look interesting.

Some books on the methods of natural language processing,
including LDA with reasonalbe amount of math left not unjustified? Im probably going to use it soon and would like to get some understanding and feeling for it beforehand. I am not sure what rhe prereqs are for sufficient in depth understanding of the topic. Assume knowledge of Linear Algebra up to jordan form, normal form, bilinear forms, analysis on manifolds, prob theory and mathematical stats (with out measure theory though).

>>12104418
You can't really say without knowing the chance the coin is fake in the first place.

>>12104418
Use the binomial test - basically find the probability of getting a less equitible result assuming the coin is fair.

>>12104595
Try Bishop, probably, depending one exactly what methods you're interested. It's not about NLP and I won't pretend to know what methods people actually use in NLP but if you're after stuff like LDA it's a good place to start.

>>12103754
just replace B with the disjoint union of B, B^2, B^3 etc

>>12102117
You are welcome. Also notice that since Catalan’s conjecture has been proven true, for every other pair of consecutive prime powers one of them must be a prime.

>>12104158
Aren't oral examinations subjected to biasing/preferential treatment by the examiner? Sounds a bit random. If you take a written test you eliminate any kind of subjective judgement, you can always contest the result.

How to prove using intuionistic natural deduction that:

(((P -> ⊥) -> P ) -> P == ~~P -> P


Its trivial using a truth table, but i have no idea how to work it out using deduction.

>>12097919
is this loss?

>>12105107
>(P -> ⊥) -> P
Doesn't everything follow from this?

>>12105170
Yes, but how can i use the deduction rules to derive ~~P if P is assumed or vice versa? LEM and the derivation, double negation is not incuuded in intuitionistic logic.

>>12105233
The only inference rule you need is modus ponens, which is part of intuitionistic logic.

>>12105107
>intuionistic natural deduction
i think -p is p->bottom in this logic

search for ~~P
https://www.cs.cornell.edu/courses/cs3110/2018sp/l/20-coq-logic/notes.v

Are lattices the highest IQ algebraic structure?

>>12105513
>they don't work exclusively with CABAs

>>12097896
What can be divided by zero?

>>12105720
/pol/ outside of /pol/ should be bannable. There's no way you actually care about your question, you're just here to post that image.

>>12105720
>Phoneposter
You have to go back.

Are visual proofs valid?

>>12098234
kill yourself
and then lurk moar

>>12105785
No, but they can help with intuition.

>>12105772
How can you determine he’s on a phone?

>>12105107
You only need to show that
((P -> ⊥) -> P
<=>
((P -> ⊥) -> ⊥

The <= direction is true by concatenation with explosion.
The other direction - I don't know the details of your setup, probably follow from considering that your assumption also holds true for P=bot and bot->bit holds

my uni doesn't offer set theory this semester, I'm so sad frens I have been studying set theory for like two months aaaahhhhhhhhhh

Also is it stupid that I'm studying beforehand

>>12105861
Why on earth would you willingly take set theory is the real question.

>>12105865
oh am I not in the cool zone for studying set theory?
is set theory not worth it?

>>12105891
It is. The other anon is just being a peepeepoopoo. Set theory is really fun, you can still study it on your own. What book are you using?

>>12105917
well that's a relief since I'm really digging the subject.
I'm using Derek Goldrei's Classic Set Theory and Introduction to set theory by Hrbacek and Jech. Also I have my proff's notes.
I think Goldrei's one is particularly good as an introduction.

>>12105861
>Also is it stupid that I'm studying beforehand
no

>>12105865
why would anybody do anything.
math is just fun - everything to the contrary is just different taste

>>12105948
I have never looked at Goldrei's book. But I do know Jech and Hrbacek's book, its pretty good. I like the chapter on comibnatorial set theory and the one on large cardinals.

>>12105758
>>12105801
Go back.

>>12105785
Yes, as long as you understand how they work. Understanding how something works != being able to write down a proof. As long as you've convinced yourself, it's good.

>>12105801
Phoneposter means someone whose posts are short and/or low quality. It is assumed that if your posts are short, then you must be posting from a phone.

>>12105012
Yes, but you can "overrule" oral examination grades as well. I actually had to do that once. Overall I think having oral exams is more of a benefit than not, even if it is suspect to preferential treatment.

And I don't think preferential treatment is eliminated in written exams completely, with a small class the teacher will know each student, and the exams aren't anonymous.

>>12104209
I'm not from eastern Europe, I don't want to say which uni/country and I'll leave it as it's a top 100 university.

>>12104210
I was perhaps a bit unclear. The first half would be _largely_ computation with known techniques, and a little bit of cleverness. I.e. it's not just a straight example with numbers switched out. Of course some thinking is required, just not as much as in the second half. The second half would be _largely_ theorems, and proofs. Of course there's some overlap. For reference as I said we have 6-7 hours to answer 5-7 questions. So the questions are rather in-depth.

>>12106001
>with a small class the teacher will know each student, and the exams aren't anonymous.
True but I think at least you have a hard copy. So it is easier to prove that you were mistreated intentionally or by mistake. But I don't know how you are going to contest oral examination since there is no proof. But I agree in some cases it may be beneficial since the examiner may ask for clarifications and so you can provide immediate feedback to convince him you know what you are talking about. Whereas with written tests if the grader is in doubt, maybe you solved the problem correctly but did your calculations wrong or maybe there is a typo etc. or maybe your way of thinking was totally wrong etc but he cannot confirm that in person so you are at his mercy.

Brainlet question (I already asked in /sqt/): Are continuous partials necessary conditions for a complex valued function to be analytic in some region [math]D \subseteq \mathbb{C}[/math]?

>>12105999
fucking retarded newfag

>>12105513
>finite product theories
>high IQ

>>12097896
I'm doing a maths degree in the UK and about to head into my final year.
I have chosen Coding and Cryptography as a module over Financial Mathematics, but I would like to pursue a career in finance (not sure what yet).
Have I fucked up or does this not matter when applying to a postgraduate finance role? Will I be trained if I manage to get a role.
Thank you!

>>12106365
why did you do that then you mong

>>12106365
>I want to pursue a career in finance, but I specialized in something other than finance
What, exactly, did he mean by this?

>>12106369
>>12106390
I had to pick the module in June, and I wasn't sure what I wanted to do with my degree back then, but now after research I think I would enjoy a career in finance the most.

>>12106454
>I think I would enjoy a career in finance the most
enjoy isn't the right word, you'll enjoy the money, but trust, not a single fucking soul enjoys working in finance

>>12106464
>but trust
but trust me*

>>12106464
>>12106468
I think it's what I would like to work in though, I'm not sure what else I would like to do with a maths degree.
Will I still be able to get a job in finance?

how do I kill myself for having failed 1st year of engineering school even though I loved math (especially algebra, even abstract algrebra)?

>>12106506
If you prefer math to engineering, switch majors to applied math. You'll still take a fair amount of pure math courses.

>>12106508
I'm 35. I don't think I have enough brain to study anything anymore. also, I was poor at the time, so math was out of the plan

>>12106464
Is it really THAT bad?

>>12106509
I might be misunderstanding your post, but you can make good money with an applied math degree. Maybe not as good as en engineer, but you'll be doing good. Some of my friends even went into software engineering with math degrees.

>>12106454
huh? you can make shitloads of money in cryptography. also, infosec is more fun than finance

>>12106515
eh, I'm already making good enough money in infosec. as I said, I loved studying math, and I feel ashamed of not having continued, not even by myself

>>12106518
>huh? you can make shitloads of money in cryptography. also, infosec is more fun than finance

I’ve not really done much coding before, aside from the required bits on basic MatLab. Wouldn’t they hire someone with a computer science background for cryptography?

>>12106594
Learn it on the job.

>>12106594
well, AFAIK, you also need to know a bit of hardware, OSes, etc. and not only "coding" if you want to build efficient cryptographic algorithms. but those jobs are for infosec rockstars, I'd guess. still, I guess you are correct: you need some background in CS.
anyway, if you want to know more, I suggest asking here: https://webchat.freenode.net/##crypto

>>12106486
Yes of course the other posters are talking nonsense. Physics backgrounds are quite common for crying out loud. Not that I’d recommend this career in particular.

Can you prove Fubini's theorem without measure theory?

>>12106808
in the discrete case yeah

I find it amazing that addition beats multiplication in only one special case: when adding ones. It is taken for granted but isn't it amazing that 1+1+1... >>>> than 1*1*1 and the difference grows as the number of dimensions goes up?

>>12106959
how about this quick fact?
Every hyperoperation between 2 and 2 always results in 4
2+2=4
2*2=4
2^2=4
2^^2=4
etc etc

>it's another summerfags and tourists ruin /mg/ thread
it never ends

>>12107217
what is ^^?

>>12107248
i unironically miss the gmmg tranny

>>12107256
tetration
now don't be lazy and go read about hyperoperations on wikipedia

>>12107248
>He thinks summer ever ended
Absolutely new.

Is it reasonable to define a 'point-like object' in a category to be an object whose covariant functor of points is naturally isomorphic to the identity functor?
The singleton satisfies this in Set/Top right?

What are some good resources for learning math as a layman?

>>12107248
I didn't do nothing I swear. I'm doing baby calculus before I wake up my gf with oral

>>12107217
bro what the fuck

>>12107340
Youtube and general google search. Don't bother with text books. ALL of them are horrible. Use them as a reference, as a list of topics to study. Then just search and download a bunch of pdfs and watch the videos. Many of them ARE based on exact same text books except they actually explain what is going on. To me it is like Torah which is the detailed explanation of the cryptic Talmud. The only useful type of text books are the Schaum's outline series since it is just a bunch of problems with solutions.

I especially struggled with the Mishnah, which is the tersest motherfucker there is. This is a good analogy to studying math using "text books". Fuck text books.

>>12107365
>Youtube and general google search. Don't bother with text books.
You have this backwards.

>>>/diy/1905770

>>12097896
First time posting.
Having an existential crisis.
>integers are arbitrary
>relationship between any perceived time/space value is some kind of functional derivation from a universal point or reference
>universe will probably return to a static state when the distortion of space/time begin to approach the universal rate of expansion from the origin point.

As a kid I could never understand time, clock time is relative to the motion of the earth, percieved time is obviously just an misrepresentation of motion and has nothing to do with time.
Actual time is a function and so can only be experinced by contrasting two arbitary points.

Can my head collapse if there's nothing in it?
The universe is going to collapse.

>>12107549
>be me
>13
>ask teacher how magnets work
>says they have a field which is positive or negative
>ask how that force is transmited
>charges particles between them or some shit
>Sir then how does the moon orbit the earth if there's nothing in space
>get told it's different and to shut up
>but sir cow come the moon effects my compass if gravity and magnetism aren't the same
>got a detention
>try to explain to the principle the seriousness and implications of the relationship between gravity and magnetism (pseudo gravity)
>get sent to see the psychologist
>stare at the psychologist, he stares at me for ten minutes silently
>tells me I can go back to class

>>12107561
>be in class
>teacher draws a graph
>ask why the graph starts at 0 and not 1
>teacher tells me to shut up
>Sir how do you even know what a value under 1 is?
>It's called a decimal value shut up
>sir could it be a fraction?
>yes .5 is 1/2 how are you this dumb anon
>But sir aren't both are half of one not half of zero, isn't 1/2 just negative .5 relative to 1?
>had to wait outside the class.

>>12107578
>be me
>yesterday
>trying to explain about the virus
>person tells me that I'm talking about some kind of calculus and have to study it before I can explain
>look up calculus again
>oh no fuck it's that fucking line thing again
>the one that I got kicked out of school for arguing with my phis teacher about.
>post on /sci
Why can I not escape the knowledge that all mathematics is reductionist of fundamental concepts we still don't understand.
Time is some kind of joke, mass is satire- our measurements of mass, time and velocity, distance are all some kind of joke.

>>12107545
Holy shit hahahahahaha

>>12107545

>>12107333
Initial object.

>>12107545
BASED

>>12107545
LOL

>>12107278
This, and yukari and work-with-physicists-anon

>>12107545
lol

it's actually crazy to me– you'd think doctors would be among our best and brightest, but at least at my uni (which places people at top med schools) the pre-med kids are by in large of mediocre intelligence, albeit quite driven.

>>12107545
so this is why they developed Tai's method

>>12107705
>Pic
Math for this feel?

>>12107256
it's a German emoji, means happy laughing

>>12107789
The umbral calculus

On Von Neumann's wiki page, it's stated that he argued for a group-theoretic interpretation of the fact that the "problem of measure" on [math]\mathbb{R}^n[/math] admits a positive solution for [math]n=1,2[/math] and a negative one (due to Banach-Tarski) for [math]n \ge 3[/math]: "the existence of a measure could be determined by looking at the properties of the transformation group of the given space [...] this comes from the fact that the Euclidean group is solvable for [math]n \le 2[/math], and unsolvable for [math]n \ge 3[/math]".

QUESTION: It's known that a random walk on [math]\mathbb{Z}^d[/math] is recurrent for [math]d=1,2[/math] and transient for [math]d \ge 3[/math]. Can /mg/ offer a similar group-theoretic perspective on this?

bros... i forgot to do maths today... i feel sad...

>>12107935
how so

>>12107978
I can't
but we can study Banach-Tarski

>>12108004
I haven't done (pure) math in 5 days, it's ok anon.

>>12106772
okay thanks, I might email the uni and change to financial maths and teach myself coding at the same time

I have an idea how a programming language that allows for arbitrary precision computation of mathematical problems in an elegant manner. It would be an intermediary between numerical computation and symbolic manipulation.
The idea is that every mathematical object is represented by a finite algorithm which computes that object to arbitrary precision (in the best case it also knows an error bound)
This idea only works for computable mathematics. So given two computable real numbers a and b, they would be represented by an algorithm whose output converges to the number when given enough time. The sum of a + b is then a combination of the two algorithms. Here it is important that some smart reductions/optimizations are made so the final algorithm after performing a few computations isn't exponentially inefficient. This would be compile time optimization maybe.
A continuous real function is represented by an algorithm that computes the output given an algorithm that computes an input. Then the integral of the function over an area is also just an algorithm which converges etc.

>>12107545
OK but did you read the entire thread? What do they mean by a non linear laplace transform to solve non-linear DEs? I am not aware of such a thing.

>>12107545
FUcking hilarious

>>12108121
Good luck getting anyone to read your code

Weird thing in complex differentiation:

>df/dz along x and y paths are equivalent
>assume f has twice differentiable partials
>travel one dy, then one dx from start point
>by pythagoras, you have traveled rt2 infinitesimal units
>since dx is approximately the same at f(0) vs f(dy), you have grown the functional value by df/dy+df/dx = 2df/dz
>in traveling rt2 units you have grown 2 df/dz units
But this is contradictory, implies the rate of change is not one df/dz per unit local change, but instead is 2/rt2 here

What gives?

>>12107691
its a terminal object. the initial object is the empty set. It's more so that the forgetful functor is representable by the point (in a covariant sense).
I'm just thinking about how to generalize what the forgetful functor should be like in a general category, by noting that in Top and Set you can 'decode' any morphism [/math]f: X \to Y$ by looing at [math] Hom( \dot , X) \to Hom(\dot, Y)[/math]

Yo, can anybody help me proceed with my Real Analysis homework? So far I have figured that I ought to take the contrapositive of the first problem, yielding
[math] \forall j \in J, \exists \alpha \in A(Y_\alpha \cap Z_j = \emptyset ) \Longrightarrow \exists \alpha \in A, \forall j \in J (Y_\alpha \cap Z_j = \emptyset )[/math]

but I don't know how J being finite helps me nail this down. I don't think I need contrapositive form for 4b.

>>12109236
theres a relatively simple counterexample for b) if you take X = \mathbb{N}

>>12109236
[math](A_1\cap A_2) \cap Z_1\subset A_1 \cap Z_1 = \emptyset[/math]

>>12109265

You are just trying to (dis)prove: for all Z_j, there exists a Y_alpha such that is disjoint from Z_i. This is the negative, for sake of proof by contradiction. now apply the assumption "if we know...". Does this lead to contradiction?.

What's the point of including the du function in an integral involving u-substitution if the derivative isn't carried into the final result?

>>12109340
example?
you sound mistaken

>>12097896
How many apples is that though?

>>12109441
A Rayman surface of apples

>>12109452
This.

Are Lebesgue measures/integrals required if you want to truly understand statistics and probability?

>>12109880
This looks retarded lol

>>12109642
yes if you want to read what other people write

>>12109642
no not at all. all job stats are on reimann integrable functions. you can know what a measure space is and the basic parlence without taking a bona fide course on measure theory and studying shit like radon measures, haar measures, non measureable sets etc

>>12108917
Infinitesimals don't exist bro jeez

>>12110209
They exist so much there are like five different formulations in common use.

>>12105513
>Are lattices the highest IQ algebraic structure?
>not quantales

wew

>>12107545
doctors are normies larping as scientists

>>12108121
equality is not decidable on R

Do I actually need all the background trigonometry has? Like I know most basic concepts like unit circle and such but I was wondering if I can just learn the formulas for calculus and do just fine?

>>12110583
For differential calculus, you'll only need the basics. For integral calculus, you'll use more like double angle formulas and such, but nothing extensive.

This question blew my mind, it seems pretty difficult but if you draw a "picture" of the underlying schemes the answer is obvious.
Let [math]A = \mathbb{C} [x,y]/(xy) [/math] and [math]B = \mathbb{C} [x,y]/(xy^2) [/math]. For which [math]f \in A[/math] and [math] g \in B [/math] are the localisations [math] A_f [/math] and [math] B_g [/math] isomorphic?

>>12110035
Convergenve in probability makes no sense without measure theory. Heck defining random variables and probability measures make no sense without it.

>>12110724
I don't know what a localization is lol

>>12110808
https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
Localising by [math] f [/math] is basically saying you are now allowed to divide by [math] f [/math]. Elements of the localisation [math] A_f [/math] are "fractions" of the form [math] \frac{a}{f^n}[/math] where [math] a \in A[/math]. Two fractions [math] \frac{a_1}{f_1} [/math] and [math] \frac{a_2}{f_2} [/math] are equivalent if there exists an [math] m \in \mathbb{N} [/math] such that [math] f^m (a_1f_2-a_2f_1)=0[/math] in [math] A [/math]. This is basically the normal equivalence for fractions except you have this extra [math] f^m [/math] term which helps you avoid dividing by zero when [math] A [/math] is not an integral domain.

>>12110280
Not in standard analysis tho.

>>12110877
Not this guy but I've come across this several times but never got an intuition for it.

Is there some low-dimensional (but higher than 1 dimensions) algebra or module or space X, maybe even a matrix algebra, with a localization where the things one localizes w.r.t. are not just 0 and the rest? E.g. some 2 or 7 element set one uses in this process. I'd like X becomes after a non-trivial example. What it's elements are and how many there are and such. thx

>>12110979
Think more of simple rings.
In [math]\mathbb{Z}[/math] for instance, you can localize with respect to 5 and you get [math]\mathbb{Z}\frac{1}{5}[/math].

Another classic exemple is you take a prime. 5 again for instance, and you localize everyone that isn't a multiple of five. Then you get [math]\mathbb{Z}_5[/math], the ring of 5-adic integers, whose only prime number is 5.

>>12110724
We the schemes they define clearly have the same underlying sets, but one is reduced and the other isn't. Now, there's only one structure of reduced scheme in general for a given underlying set, so any localisation of the second set that is reduced will work.
This means we have to eliminate xy from the second ring.
So any localisations [math]A_f[/math] and [math]B_g[/math] where f is the image of g in [math]B/(xy)[/math] and g is a multiple of y.

Is that correct?

>>12111039
>Z(1/5)
What's that?

>localize everyone that isn't a multiple of five
this means only those elements survive in a quotient?
I don't know much about the p-adics

>>12111039
>Then you get \mathbb{Z}_5, the ring of 5-adic integers
I'm baked as fuck but I'm pretty sure this is wrong, the same notation is used but those rings are different (-1 has square roots in the p-adics for example).

>So any localisations AfAf and BgBg where f is the image of g in B/(xy)B/(xy) and g is a multiple of y
That's basically the tricky part, there are more that come from scaling based change of coordinates though.

>>12111132
>What's that?
The integers but you can divide by 5 :^)

>this means only those elements survive in a quotient?
The integers but you can divide by everything that isn't a multiple of 5 :^)
Like the rationals but you can't divide by 5

>>12111136
>Like the rationals but you can't divide by 5
So 5 is not in it anymore (or, if it is still in it, I can't divide 5 by 5 anymore)?

I'm trying to phrase the uncountability of PN in second order arithmetic - can you verify my thinking...

Naturally, I phrase it as non-existence of a bijection between N and (N->{0,1}), which we can uncurry to (N x N) -> {0,1}.
We can speak about those as functions by applying a pairing n->(k,m) twice (those functions are encoded as subsets of N and thus we're still second order).
Now I just need to express the bijection property.

A bijectionN N->{0,1}) as encoded by ((N x N) x {0,1}) means

for all m,n, if
??? profit ???
then m=n
I still need to figure out the ??? part in terms of triples

[I couldn't find anything on Cantor diagonalization without set theory]

>>12111142
5 is in
1/5 is not
43/6 is in
36/10 is not

>>12111145
Okay thanks. In what sense is this a "localization"? Seems like you removed info about 5.

>>12111147
You probably have no idea what a scheme is but Spec [math]A_f[/math] embeds as an open set in Spec [math]A[/math] so it is a "local" piece of information. Localising at a prime is the same as the stalk at that prime, which is even more "local" information.
Often when localising you get a local ring (https://en.wikipedia.org/wiki/Local_ring)) but I'm not sure what terminology came first.
>Seems like you removed info about 5
In fact what you're really doing is removing information about all the ideals except (5) (and powers of 5). For example since 4 is invertible the ideal (4) becomes the trivial ideal (1). You add more elements but the ring becomes simpler as a result.

>>12111147
5 is the only remaining prime ideal, so you can study how some equation relates to 5 as a prime without having other primes getting in the way.

>>12111136
Ah yes, I somehow forgot. You get the 5-adics as the completion of [math]\mathbb{Z}_5[/math], not directly as the localisation.>>12111136

No, you're right, I talked way too fast.
You'd still have to take the completion of that [math]Z_5[/math] to get the 5-adic numbers.

>>12111165
>>12111229
Okay, so it's a kind of trivialization ("field-ification") of everything else

>>12111230
Exactly

>>12097896

>>12111230
Yes! If we consider an integral domain, then the trivial ideal is prime. We may now localise with respect to that and this is precisely the field of fractions.

>>12106288
Yes. It is even true that both the real and imaginary part of a holomorphic function are smooth. Should be somewhere in any introductory book to complex analysis

>>12111257
okay, good to know. maybe I read some AG sometime

>>12111292
Just your basic ring theory, but if you wish to walk the AG path I will not stop you.

>>12109014
https://ncatlab.org/nlab/show/generalized+element

>>12108917
>infinitesimal
stopped reading there

>>12111290
Thank you.

1466282

>>12111249
Why are americans so fucking insane

>>12112291
Neue

[math]\mathbb{Q}[/math] is a metric space, right?

>>12112329
Yes

Just did my first quiz for discrete math and got a shitty grade. Is this class hard or I'm just retarded?