/mg/ maths general

[Anonymous No.12224197]

Talk maths.
Previously: >>12214123

[Anonymous No.12224212]

>>12224197
>jennifer lawrence
literally who?

[Anonymous No.12224242]

>>12224197
I did some group theory years ago for my nm thesis and I've never seen the notation in the bottom right.

[Anonymous No.12224251]

>>12224212
He means Jennifer Lopez.

[Anonymous No.12224266]

I'm been playing with lattices and here's a quest 4u
(haven't solved it yet, but I suppose the number is divisible by 3 or something)

We consider the commutative semirings (R, ·, +, 0, 1) which are idempotent (x·x=x), simple (x+1=1).
(The idempotent relation makes this lattice-operation like and the simple condition makes this bounded-like (1 behaves like the maximum). The distributive properties of the semiring make those into the distributive lattices of that kind)

Let's pin it down to a semiring of exactly five elements, with different elements 0, P, Q, R, 1.
The identity properties for the semiring for 0 and 1, and the simple property, imply most of the multiplication and addition table.
The question is what simultaneous assignments for the six remaining operations
P·Q, P·R, R·Q,
P+Q, P+R, R+Q
is possible.

Notes:
I've looked at the situation with four elements 0, P, Q, 1 already.
You can define an order (x<=y) := (x·y=x), which is equivalent to (x<=y) := (x+y=y).
Then if you assume 0<P,Q<1, the multiplication table is fixed. This will also work for the situation with five elements. But there's also "diamonds" of lattices and such.

Relevant links
https://en.wikipedia.org/wiki/Lattice_(order)#Bounded_lattice
https://en.wikipedia.org/wiki/Semiring

[Anonymous No.12224284]

maybe here as the multiplication table for the 6 missing entries

The thing is commutative, so the tables are symmetric for both multiplication and addition.
And we have
x + 0 = 0
x + 1 = 1
x · 0 = 0
x · 1 = 1
x · (y+z) = (x · y) + (x · z)
as distributive laws.
I think that's all.

[Anonymous No.12224290]

Maybe here as the multiplication table for the 6 missing entries

The thing is commutative, so the tables are symmetric for both multiplication and addition. And we have

x + 0 = 0
x + 1 = 1
x · 0 = 0
x · 1 = x
x · (y + z) = (x · y) + (x · z)

I think that's all.

How does this then go on for size n semirings?
Does every size k semiring inject into one of size k+1 with the same properties?

[Anonymous No.12224299]

0<P<Q<1

[Anonymous No.12224336]

Consider the presheaf of continuous complex-valued functions [math]\mathscr{C}[/math] on a circle [math]S^{1}[/math] and the presheaf [math]\mathscr{C}^{*}[/math] of continuous complex-valued functions that do not vanish anywhere along the circle; the first is a sheaf of abelian groups (under [math]+[/math]), and so is the second (under [math]\cdot[/math]). Use the exponential map to define a morphism [math]\text{exp}: \mathscr{C}\to \mathscr{C}^{*}[/math] .
Prove that [math]\text{exp}[/math] is not an epimorphism of presheaves: show that its cokernel has value [math]0[/math] over every proper open set of [math]S^{1}[/math] but is nonzero over [math]S^{1}[/math].

[Anonymous No.12224342]

>>12224336
Does this involve much topology? Maybe instead of speaking of a presheaf, you should just write down the function spaces.

[Anonymous No.12224371]

Do you guys do this kind of stuff? Particularly the predicting the proofs of theorems, or trying to sketch them before reading them.

[Anonymous No.12224379]

>>12224371
yes, especially in algebra.

[Anonymous No.12224406]

>>12224242
What's your first language? That's cycle notation in a permutation group.

[Anonymous No.12224416]

>>12224336
I did a variant of this question in my algebraic geometry course! Basically the point is that locally logarithms exist (on simply connected opens), but there is no global logarithm on the circle (insert complex analysis here).

[Anonymous No.12224419]

Any ops management bros here?
I am working on Quantity Discount Model problems and have a question that my chink professor won’t answer:
If I find out that the minimum number of orders needed to meet demand is a fraction (say, 600 total demand, 49 units per order) do I use 600/49 in my subsequent calculations or do I round 600/49 up to 13?
Logically I would think that you can’t have 12.24 orders so I think I need to round up but I don’t quite know

[Anonymous No.12224424]

>>12224379
I know it's not the same but I asked precisely because today I tried this with my Linear Algebra book. It was actually kind of fun, although some of the proofs were a bit beyond me.
I couldn't imagine doing it for Calculus/Analysis though.

[Anonymous No.12224431]

>>12224419
>chink
Why the racism anon?

[Anonymous No.12224435]

>>12224242
<g> = {g^k, k an integer}
(1,5) is the permutation that sends 1,2,3,4,5 -> 5,2,3,4,1
S_5 is the group of bijections from a set of 5 elements to itself

[Anonymous No.12224458]

brain is not working today, how can i show that the series is divergent for 1>p>2 ?
i know that it's divergent for p=1 and convergent for p>=2

[Anonymous No.12224496]

>>12224458
Cauchy condensation.
2^(n(1-p/2)) dominates n^(p).
Also
>1>p>2
lol

[Anonymous No.12224507]

>>12224496
ye but isn't n^(-p) convergent for p>1 ?
>1>p>2
im tired

[Anonymous No.12224515]

>>12224507
Yes it is, but that doesn't matter. You get the sum over n of
2^(n(1-p/2))/n^p.
As I said, the fact that the numerator dominates the denominator shows that the series must diverge.

[Anonymous No.12224521]

>>12224515
ah yep makes sense thanks anon

[Anonymous No.12224527]

>>12224521
Don't mention it.

[Anonymous No.12224654]

>>12224212
Harvey Weinstein's cock sleeve.

[Anonymous No.12224691]

Anyone here like computability theory?

[Anonymous No.12224726]

Give me some examples of a function on [math] \mathbb{R} _+[/math] that's strictly positive, concave, and decreasing. I can't think of any simple but interesting ones.

[Anonymous No.12224734]

>>12224726
[math]f(x)= x^{-1}[/math]

[Anonymous No.12224875]

>>12224691
Yes, I love it! Haven't really done any in like a year though.

[Anonymous No.12225136]

>>12224734
>concave
>>12224726
i doubt such a function exists, the instant it decreases even a little it must keep up that rate of decrease at least in the future, inevitably hitting 0.

[Anonymous No.12225416]

If you're such geniuses as you claim to be, why haven't you solved the Navier-Stokes smoothness problem and claimed the 1 million dollar cash prize?

[Anonymous No.12225443]

>>12225416
That would require actually helping people. I'd rather wank to abstract nonsense personally.

t. Future Algebraic Topologist

[Anonymous No.12225445]

Got a test tomorrow consisting of mathematical physics problems. They're probably going to ask me to solve a few integrals using complex variables, solve Laplace's equation a few different ways, and work with some bilinear maps.

Did bad on some of the assignments and I've been studying them and the solutions but I'm still nervous. Any advice?

[Anonymous No.12225450]

>>12225443
It's a pretty abstract problem to be fair

[Anonymous No.12225481]

What kind of job (outside academia) could you get after a second year master in optmization? I'm planning on doing this https://www.imo.universite-paris-saclay.fr/-optimization-?lang=fr
Should I prepare a thesis after?

[Anonymous No.12225507]

>>12225450
This. No physicist or engineer cares about muh existence and smoothness. They are empirically so, and that's enough. Work on making their computation more efficient instead.

[Anonymous No.12225599]

I'm trying to remember this interesting problem I learned about a while ago but I can't recall enough details for google to be helpful. All I remember is that you drop some sticks on the ground and their positioning turns out to be related to pi in some way.

[Anonymous No.12225643]

>>12225507
>This. No physicist or engineer cares about muh existence and smoothness.
That's why it's math fag problem that still needs to be solved by math fags.

[Anonymous No.12225731]

>>12225599
I think d'Alembert came up with this one

[Anonymous No.12225736]

>>12225599
>>12225731
my bad, this is known as Buffon's needle problem

[Anonymous No.12226013]

>>12225136
>it must keep up that rate of decrease at least in the future
That doesn't seem to follow. At least in an interval, it could be concave throughout but the flow f'(x) actually increases. I.e. it becomes less concave. Or does this imediatenly lead to an inflection point towards being convex?

[Anonymous No.12226219]

>>12226013
If the derivative decreases, it's not concave there. You can have the absolute value of the derivative decreasing, if it's negative, but that only works in one direction. e.g ln(x) is concave but only stays positive in one direction. Or ln(1/((e^x) +1)) +50 if you want something defined on the whole line.

[Anonymous No.12226342]

>>12226013
>>12226219
Wait wait, I mean if the derivative *increases* it's not convex there, not if it decreases. Got a bunch of signs wrong, but the examples still work.

[Anonymous No.12226620]

Really struggling in my stochastic processes course, anyone have some good resources to share? Youtube or whatever? The lectures and book are quite difficult because they're just
Definition
Theorem
Proof
over and over again.

[Anonymous No.12226696]

>>12226620
>Definition
>Theorem
>Proof
>over and over again.
Get used to it.

[Anonymous No.12226713]

>>12226696
Oh ok thanks, really exhaustive reply

[Anonymous No.12226757]

>>12226620
What kind of retard complains that the course is too structured?
If you don't like it just erase the definiton theorem proof and read it as a smooth text.

[Anonymous No.12226827]

>>12226757
Yes I'm kind of dumb, thanks for the awesome links btw
There is no smooth text, that was the point of the fucking question

[Anonymous No.12226856]

C^inf brain is my favourite insult

[Anonymous No.12226882]

I need a function that looks kinda like this on [0,1] but is *not* a translation of another function. It needs to be concave. Any ideas?

[Anonymous No.12226883]

>>12226882
>but is *not* a translation of another function
Literally every function is a translation of another function.

[Anonymous No.12226981]

Does anyone know where I could find the movie "The Colors of Math"?

[Anonymous No.12227228]

>>12226856
At what angle does it determinate?

[Anonymous No.12227236]

>>12227228
>Anonymous 10/13/20(Tue)14:21:01 No.12227229▶
wut? C^inf is smooth.

[Anonymous No.12227255]

>>12227236
By C do you mean a complex number? If you have a complex number self multiplied infinite times, it would rotate itself infinite times, but what would it land on?

[Anonymous No.12227266]

Say I have a list of numbers, {a1,a2,a3,...}. I know that each of them is an integer multiple of some constant x, {b1x,b2x,b3x,...}. I don't know what the b's are or what x is, is there any way that I can figure out the value of x?

[Anonymous No.12227291]

just bombed a maths exam, feels awful bros. homeworks were all a breeze but I just wasn't quick enough today. not sure why professors give exams of the same difficulty as homeworks when you have one hour to do the former and a week to do the latter :(

[Anonymous No.12227308]

>>12227266
You take the greatest common divisor of a_1, a_2, ...
Then you know x must divide this number. You can't figure out anything else. Unless the gcd is 1, x is undetermined.

[Anonymous No.12227324]

>>12227308
My issue is that my spreadsheet software is rounding the values to integers

[Anonymous No.12227412]

>>12226713
>>12226827
While I understand that you don't enjoy the nastiness of 4chan responses (though I wonder what you expected), the truth is you didn't give very much detail yourself.

What exactly are you struggling with?

Are you struggling with just the act of reading the definitions, theorems, and proofs? Following along at all? If so, then "get used to it" isn't really awful advice...you should go work some text that will make you comfortable with rigor. But it is hard to suggest what that should be, since we don't even know your background or what you have already tried (if anything). This is a failure on your part -- provide more information next time.

Are you struggling with specific concepts in the course itself? If so, then you are right maybe some links or commentary in the responses would be helpful. But it is hard to give this sort of response if we don't know what concepts you are struggling with. This is a failure on your part -- provide more information next time.

[Anonymous No.12227468]

This is kind of a silly question, but what is "the point" in picking contours for complex integration? Obviously if you can compute it, one contour is as good as another, but do you guys have anything you look for?

[Anonymous No.12227472]

>>12227255
>By C do you mean a complex number? If you have a complex number self multiplied infinite times, it would rotate itself infinite times, but what would it land on?
who the fuck notates a complex number by C? Think it's pretty standard to use z.

C^inf refers to the class of functions that are infinitely differentiable and therefore in a sense "infinitely smooth"

[Anonymous No.12227494]

>>12226713
>>12226827
>>12226620
When in doubt, you can always try to use MIT OCW. Check the lecture notes from one of these courses, try to find the topic you want to learn more about.
https://ocw.mit.edu/courses/mathematics/18-445-introduction-to-stochastic-processes-spring-2015/
https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/

More importantly, those courses have posted problem sets with solutions. That is very valuable as the best way to learn is by solving problems and working through examples.

You need to resist the feeling of frustration; it's often difficult to see the big picture when studying textbooks or listening to lectures. So you often don't easily see why a new definition is motivated or why a new theorem is interesting. You need to work with the concepts yourself, and that means solving problems and working through example exercises. Try making up your own examples for every abstract definition. Good luck with your course.

[Anonymous No.12227552]

>>12227472
You should just have called him a [math]C^\infty[/math]-brain for that comment, dude.

[Anonymous No.12227614]

>>12226620
>The lectures and book are quite difficult because they're just
Try working on the exercises.

[Anonymous No.12227655]

Holy shit it all makes sense now.

[Anonymous No.12227662]

>Book explains hypothesis/postulate/etc
>Write it down
>realize I misunderstood it but the method of the author is arguably less practical and arbitrary but there is a chance other people are going to see it

Do I just write down both methods and explain why mine is better?

[Anonymous No.12227664]

>>12227468
The point of picking it is to allow you to compute in the first place, because how could you do an integral without it? The point of one over another is simply preference/ease of use as you supposed

[Anonymous No.12227669]

>>12226620
Spend time visualizing and developing intuitions for the relevant definitions and theorems.

[Anonymous No.12227699]

>>12224371
yes i do it and i think everyone should do it
I see a new theorem -> i think about the proof
I have some ideas -> fuck around and try to prove it for a while
I don't even have a clue where to start -> read the beginning of the proof, then try again
In the case i can't prove the theorem myself, i "read" the proof as in "verify that every next sentence follows from the previous sentences" and then i contemplate until i understand better. By this i mean: even the tricky proofs contain one or two smart ideas and the rest is details. you need to extract the key idea, once i understand the proof well, i'll be able to say something like "oh so i just define ABC and then observe that XYZ holds and the rest is obvious"

obviously often im too lazy but i think that's how a proper reading of a mathematical text looks like

[Anonymous No.12227712]

>>12227699
Are you the finnish bro? How are you doing man?

[Anonymous No.12227732]

>>12227712
im not finnish, i think u've got the wrong person

[Anonymous No.12227746]

>>12227732
Are you a tranny? If so kys.

[Anonymous No.12227748]

>>12227746
im not
id like to fuck a tranny though

[Anonymous No.12227774]

>>12227748
Be honest with yourself. You want to fuck anything that resembles a beautiful woman, regardless of whether or not she's trans.
Which country are you from?

[Anonymous No.12227777]

>>12227774
Finland

[Anonymous No.12227783]

>>12227777
checked

[Anonymous No.12227798]

>>12227777
Tampere?

[Anonymous No.12227806]

>ask genuine question to prof
>tells me a statement
>Given A and B are subsets of S: A - B = A - (A union B)
>ask how to prove rigorously
>says it s obvious
And this is why I am glad i am making 100k in a cyber security firm in Texas out of college.

[Anonymous No.12227811]

Can someone explain this to me.

2-6+6x8 = ?

I think I might be retarded, because I cannot understand the logic behind the outcome.

I am teaching myself arithmetic and do not fully understand all of the fundamental laws yet.

[Anonymous No.12227818]

>>12227806
Actually it just clicked

[Anonymous No.12227821]

>>12227712
That would be me.

[Anonymous No.12227822]

>>12227811
Oh fuck, never mind I understand what's happening.

That's embarrassing.

[Anonymous No.12227823]

>>12227777
Checked and kekked

[Anonymous No.12227842]

>>12227823
That's not funny!
>>12227821
Are you GoatseHitler?

[Anonymous No.12227858]

>get amped up on coffee
>ready to finally solve the problem I'm stuck on
>check 4chan real quick
>suddenly it's 2 AM

[Anonymous No.12227862]

>>12227858
>coffee

[Anonymous No.12227892]

>>12227842
Nope, sorry.

[Anonymous No.12227908]

>>12224242
i've literally done 2.5 bare minimum algebra core courses (prefer analysis) and I know cycle notation, its literally building block stuff.

idk how you have never seen it if you studied up to graduate level algebra

[Anonymous No.12227971]

Guys I just can’t understand conditional expectation. The notation make it so hard to understand. I might drop out just because I just can’t understand a single little notion. I tried multiple resources, multiple books or pdfs but it’s just fucking terse with no meaning in my head. I tried the measure theory version and the baby first steps in probability but it’s the same non sense to me. I hate this feeling so much.

[Anonymous No.12228078]

>>12227892
No I think you are that person. Not too many representatives of your nationality here.

[Anonymous No.12228134]

Say I have vector spaces V,W,Z with an injective linear map [math]t: V\to W[/math], a surjective linear map [math] s: W\to Z[/math], and a linear map [math] u: Z\to W[/math] such that [math] su = 1_Z[/math], why is there a linear map [math] v: W\to V[/math] such that [math] vt = 1_V [/math]?

[Anonymous No.12228149]

>>12228134
Because assuming the axiom of choice, any linear map from a subspace can be extended to a linear map from the whole space, because any lin independent set can be extended to a basis.

[Anonymous No.12228179]

>>12228078
I'm also Finnish.

[Anonymous No.12228193]

>>12228179
Are you GoatseHitler then?

[Anonymous No.12228195]

>>12228179
How much of this is true?

[Anonymous No.12228221]

>>12228193
No.

>>12228195
Can't vouch for the lotto ticket, but the bucket thing is partially true. Free buckets were a part of a corporation's promotional campaign when they were absorbing a bunch of other companies under their brand, so each time a place would reopen under the new brand, there'd be a bucket giveaway. This kept going on and now it's a 'tradition'.

[Anonymous No.12228234]

>>12228078
Nope.

>>12228195
The buckets and also Lidl sneakers.

>>12228179
Ebin.

[Anonymous No.12228339]

Any good book on ring theory?

[Anonymous No.12228360]

>>12228339
Matsumura

[Anonymous No.12228390]

>>12228360
Does it covers non commutative rings?

[Anonymous No.12228476]

Given a number n, there is a probability dustribution over degree n polynomials given by flipping n +1 coins to determine whether the ith coefficient is +1 or - 1, for i = 0 to n. Then the polynomial will have n roots (counted with multiplicity) in the complex plane, so we get a distribution over complex numbers by picking a polynomial, then picking roots with probability 1/n, (Or multiplicity/n if it's a multiple root). This gives rise to some interesting fractal patterns (see https://math.ucr.edu/home/baez/roots/ ). Each of the distributions for fixed n is computably sampleable. I'm wondering, do they converge in distribution, and if so is the limit also computably sampleable?

[Anonymous No.12228584]

Let me repeat the question here.
A closed connected topological 2-manifold in R^3 is such that its orthogonal projection onto any given plane is a disk (in the rigid sense). Does it follow that the manifold is a sphere(in a rigid sense)?

[Anonymous No.12228608]

>>12228584
Are the disks the same size?

[Anonymous No.12228614]

>>12228584
Yes. You should be able to prove this. Reason about the possible combinations of prohections and and try to rule out some possibilities. Note that projections cannot increase distances i.e. if the projections of two points are have a distance of d then the actual points have a distance >= d.

[Anonymous No.12228631]

>>12228608
Exercise: prove that in fact they must be, i.e. if every projection of a shape is a disk, all the disks are the same size. It shouldn't be that hard.

[Anonymous No.12228784]

>>12227971

[Anonymous No.12228835]

>>12224212
celebrity who is commonly mistaken for various representations of cyclic groups of order two.

[Anonymous No.12228904]

>>12228835
>commonly mistaken for various representations of cyclic groups of order two.
I hate when this happens.

[Anonymous No.12228916]

>>12227971
"what is probability of thing happening if you already know other thing happen"

[Anonymous No.12228935]

>>12227806
Only true when A is a subset of B. Did you mean A - B = A - (B - A)?

[Anonymous No.12229001]

>>12228584
Can't you get a counterexample where the torus is a handle body with the handle on the «inside»?

[Anonymous No.12229278]

http://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf

[Anonymous No.12229518]

>>12224242
>I did some group theory years ago for my nm thesis and I never saw cycle notation of permutations (by implication never used GAP).
fixed it for you.

[Anonymous No.12229586]

Besides Burnside and Orbit-Stabilizer theorem, are there any other connections between the number of orbits and and the size of the group acting in a group action?

[Anonymous No.12229629]

>>12229586
The Class Equation?

[Anonymous No.12229888]

>>12229001
You'd have to cut off bits of the sphere to attach it and that would leave dents in the projected disks

[Anonymous No.12229891]

>>12224197
Maybe no one here deals with type theory/hott but I don't get how to show that two functions are equal in the sense that their equality type is inhabited (not judgmental equality).
Basically, say [math] f, g : \Pi_{x:A}C(x)[/math] and I have a function which proves equality of their values for every input, so [math] p: \Pi_{x,y:A} f(x) =_{C(x)} g(x) [/math].
Then shouldn't I be able to demonstrate a [math] q: f =_{\Pi_{x:A} C(x)} g[/math]?
I feel like there should be a function which does this:
[eqn] equal: \Pi_{f,g : \Pi_{x:A} C(x)} \left( \Pi_{x,y:A} f(x) =_{C(x)} g(x) \right) \longrightarrow f =_{\Pi_{x:A} C(x)} g [/eqn].

[Anonymous No.12229932]

>>12224266
>>12224290
So does nobody have any input?

>>12229891
In what context do you even have to prove this?

What you want is "function extensionality" and apriori it's not a feature of each of Martin Löf's type theories. In HoTT iirc it follows.

Of course they main/famous axiom to prove equality of two types is univalence, saying that A==B -> (A=B) where by == I mean type equivalence, which is when you get find functons (equivalences) putting A and B as images of another

[Anonymous No.12229936]

>>12224266
>>12224290
So does nobody have any input?

>>12229891
In what context do you even have to prove this?

What you want is "function extensionality" and apriori it's not a feature of each of Martin Löf's type theories. In HoTT iirc it follows.

https://ncatlab.org/homotopytypetheory/show/function+extensionality
https://ncatlab.org/nlab/show/function+extensionality
Hopefully there's a longer/proper article on this, but that's the keyword anyway

Of course they main/famous axiom to prove equality of two types is univalence, saying that A==B -> (A=B) where by == I mean type equivalence, which is when you get find functons (equivalences) putting A and B as images of another

[Anonymous No.12229979]

>>12229936
thanks, so basically this works as an axiom of Hott? It feels like they should've written this down as soon as they introduce identity types, it seems like the obvious question.
Is there also an axiom for cartesian product extensionality? Same situation:
[eqn]
\Pi_{f,g : \Sigma_{x:A} C(x)} \left(p_1 f =_A p_1 g + p_2 f =_{C(x)} p_2 g\right) \longrightarrow f =_{ \Sigma_{x:A} C(x)} g.
[/eqn]

[Anonymous No.12229992]

>>12228614
You just mentioned an obvious fact without explaining how it's relevant to the solution. None of what you said helps. Do you even know how to solve it?

[Anonymous No.12230014]

>>12229979
Who is "they"?

I never did much formall MLTT, but I assume on the informal level, "they" would give a quite "proof" involving the universal property of the thing.

In this case, the defining property of the pair to do exactly that can maybe be proven indpendently of you dealing with functions f and g there, i.e. just prove it for pairs (x,y)=(a,b) => x=a and y=b.
For set theory, this is e.g. exactly what the Wikipedia article on the ordered pair tries to establish as soon as they introduce a model of the pair (see bottom of pic)

The property of univalence is only a 10 year old idea or so. The MLTT rule for equality is I think still there and is what lets one otherwise deal with equality (it looks like induction, but uglier).
Not all type theories have extensionality, but I think they are glad that they have it.
(It makes it so the (n+2)^2 and n^2+4n+4 are necessarily the same function, and so is the function that takes n, generates a random sudoku, solves it, and only then returns n^2+4n+4. Although the context in which intensionality/extensionality comes into play is generally not complexity theory.)

Semi-related
Functions and extensionality has also some funny properties like the following:
Consider some proposition P and define sets
A = {x in {0,1} | x=0 or (x=1 and P)}
B = {x in {0,1} | x=1 or (x=0 and P)}
Then if P, then A=B={0,1}. If not P, then A={0} and B={1}. In either case, 0 is in A and 1 is in B.
But we don't know the truth value of P.

Now funny enough, assuming the Axiom of Choice for this (sub)finite set {A,B} implies LEM.
To see this, try to find a function
with
f : {A,B} -> {0, 1}
and
f(v) in v

[Anonymous No.12230026]

>>12229979
Unfortunately it doesn't quite work as an axiom of hott because it breaks the ability to compute with it. Some people are looking into ways to patch that but mostly they just do without it. I think you can prove extensionality for finite products but not infinite ones, or something like that.

[Anonymous No.12230070]

>>12229992
Have you even tried anything yet? Where did you get stuck? Ok, lets start simple. Choose an arbitrary plane and project onto it. The projection is a disk, call its diameter d. Any two opposite points on the circle correspond to points on the manifold with distance at least d. Are there other planes that go through those two points? What does this tell you about the projections onto those planes? Can you reach any plane through multiple steps of this? Once you've proved that the projection onto any plane has diameter at least d, remember that you could have started with any plane. Next you prove that the disks have the same center, then that this means the manifold contains the sphere with radius d/2 and that center, then that the sphere can't be extended to a larger connected manifold. If you get stuck on those last parts, let me know, but convince me that you put actual effort into it.

[Anonymous No.12230080]

>>12230014
(by function I mean set theoretical function, i.e. decided set of pairs, i.e. total function)

[Anonymous No.12230107]

G D M G

Maybe someone will like one of these
>The homotopy-invariance of constructible sheaves of spaces
https://arxiv.org/pdf/2010.06473.pdf
>On the nonabelian cohomology with coefficients in a crossed module
https://arxiv.org/pdf/2010.06493.pdf
>Up to a double cover, every regular connected graph is isomorphic to a Schreier graph
https://arxiv.org/pdf/2010.06431.pdf
>Representing Semigroups on Etale Groupoid Bundles
https://arxiv.org/pdf/2010.04961.pdf

>>12228390
If you want those, I would recommend Lam's book on those https://www.springer.com/gp/book/9780387951836

[Anonymous No.12230115]

>>12229888
How? You're taking the orthogonal projection of the whole manifold onto a plane. I can't picture a plane where that projection is not a circle.

[Anonymous No.12230133]

>>12230115
Imagine poking the hole with your finger, and choose a plane that contains the axis of your finger. The circle will be dented where your finger touches it, because it has to bend into itself to make the hole.

[Anonymous No.12230149]

>>12230107
Good day babe. Seconding Lam.

[Anonymous No.12230183]

>>12230133
I feel like that hole is filled by the image of another part of the manifold when you project.

[Anonymous No.12230213]

>>12230183
It can't be. If you take a small slice of out the edge of the circle, the only parts of the sphere that project to it are a small spherical cap. You have to take some spherical cap off to attach the handle to, and from an angle where that cap is on the edge of a circle, there aren't any points outside that cap that you could add to fill in the circle, without going outside the sphere.

[Anonymous No.12230223]

>>12230026
Wrong. You can derive funext out of univalence as soon as you also have definitional η rules (which is the case in all flavours of HoTT I know of). See e.g.

https://homotopytypetheory.org/2014/02/17/another-proof-that-univalence-implies-function-extensionality/

https://homotopytypetheory.org/2011/12/19/strong-funext-from-weak/

[Anonymous No.12230231]

>>12229979
Cartesian product extensionality holds in MLTT, you don't have to use univalence for that. It is a consequence of dependent elimination. Also, your statement is wrong, the argument is not a sum but a product, and you need to make it dependent over the proof of equality of the first component otherwise the second component is ill-typed.

[Anonymous No.12230235]

>>12230213
I think I believe you now, but I'm still not sure any such manifold has to be a sphere.

[Anonymous No.12230242]

>>12230223
Obviously univalence is sufficient, but there's plenty of models of hott that don't satisfy univalence, right?

[Anonymous No.12230248]

>>12230242
Erm, what separates HoTT from MLTT if you don't have univalence? Have you been brainwashed by algebraic geometers?

[Anonymous No.12230262]

>>12230248
are identity types part of MLTT or exclusively HOTT

[Anonymous No.12230263]

>>12230242
Indeed, here is a model of MLTT that enjoys definitional η-rules but negates function extensionality. Essentially [math]1 \rightarrow 1[/math] has an arbitrary number of elements in this model, so you can't even prove funext for finite products in MLTT.

https://hal.inria.fr/hal-01840643/document

[Anonymous No.12230271]

>>12230262
They are part of MLTT. HoTT is just MLTT + univalence axiom. (Actually, this is an abuse, HoTT is more the name research program, the proper type theory would we Cubical TT.)

[Anonymous No.12230287]

>>12230248
I am more used to MLTT than HoTT, but isn't HoTT supposed to be compatible with classical logic and univalence supposed to not be? I thought HoTT just meant you could build interesting homotopy types with something like higher inductive types and axioms to make them not collapse, or something.

[Anonymous No.12230294]

HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT HoTT MLTT?
HoTT MLTT.... MLTT!!!!

[Anonymous No.12230314]

>>12230294
OTT is the best type of MLTT anyway.

[Anonymous No.12230321]

>>12230287
>HoTT supposed to be compatible with classical logic and

It depends what you call "classical logic", which is the whole trick. Univalence is incompatible with "strong" classical logic, i.e.

Π (A : Type). A + ¬A.

But it is compatible with "irrelevant" classical logic, i.e.

Π (A : hProp). A + ¬A.

That is, you can have classical logic only over mere propositions. As soon as you have proof-relevant types, you can't be classical anymore.

> I thought HoTT just meant you could build interesting homotopy types

Yes, usually in HoTT itt is standard practice to throw in higher inductive types to create interesting new homotopy types that are not available in MLTT. But a lot of stuff is still nebulous, it's not clear what holds in the intended cubical model. Anyways, HoTT is a strict extension of MLTT. The latter is agnostic w.r.t. the status of identity proofs (i.e. it could have UIP, it could have univalence, etc.).

[Homorapist No.12230328]

>>12230271
>HoTT is more the name research program, the proper type theory would we Cubical TT.
>HoTT
>Cubical TT
>Type Theorists
NOOOOOO NOT THE RESERCH PROGRAMARINO it's called UNIVAlENT FAUNDESHONS
Fuck Type theorist
Say it one more time, I dare you, I'm gonna off myself if you do

[Anonymous No.12230336]

>>12230328
>Say it one more time, I dare you, I'm gonna off myself if you do

Wat? It's just an array of numbers in the end.

[Anonymous No.12230613]

>>12230107
I appreciate your posts even if I never read the papers.

[Anonymous No.12230649]

>>12229278
Cute
I like proofs from the book better tho, although I know it's different

[Anonymous No.12230684]

>>12230328
What are you even on about?

[Anonymous No.12230734]

Fuck reading anything about type theory is such a cock tease, I want a fuckinig database of formal mathematics NOW, I don't care if it's MLTT DTT HoTT, extensional or intensional I just want them TTs.

[Anonymous No.12230754]

Let's say you found an alien archice in the artic region buried under ahundred meters of ice which, after years of analysis, turns out to contain a digital database of mathematical statements, formalized in some system. It contains
a) The truth value of virtually any proposition you can think of
b) The truth value AND a proof.
How much would this (case a or b) be worth in terms of actual dollars? Does simply nowing the truth value of a hypothesis advance the human race?

[Anonymous No.12230765]

>>12230754
In that case I would think of the continuum hypothesis and render it non-existent. The answer is therefore: worthless.

[Anonymous No.12230766]

Does anyone know of a simple way to mentally compute the greatest common factor?

[Anonymous No.12230777]

>>12230765
> I would think of the continuum hypothesis and render it non-existent
Why? We still haven't solved the problem, it would be interesting to see the solution.

[Anonymous No.12230779]

>>12230777
Adding it as an axiom is not a solution.

[Anonymous No.12230787]

>>12230777
>We still haven't solved the problem
I'm pretty sure this was done in like the 1970s.

[Anonymous No.12230791]

>>12230684
Vovo sperged out at the end that the type theorists subverted his program. He even wrote an angry Wikipedia article distancing himself


https://en.wikipedia.org/wiki/Univalent_foundations#History

>>12230754
>found an alien archice in the artic region buried under ahundred meters of ice which, after years of analysis, turns out to contain a digital database of mathematical statements, formalized in some system. It contains
>it contains
>The truth value AND a proof.

"There exists a well-ordering on the powerset of the powerset of the powerset of the powerset of [math]{\mathbb R}[/math]"
"Proof: Axiom Of Choice."
wow, thx

[Anonymous No.12230793]

>>12230684
Vovo sperged out at the end that the type theorists subverted his program.
He even wrote an angry Wikipedia article distancing himself

https://en.wikipedia.org/wiki/Univalent_foundations#History

>>12230754
>found an alien archice in the artic region buried under ahundred meters of ice which, after years of analysis, turns out to contain a digital database of mathematical statements, formalized in some system. It contains
>it contains
>The truth value AND a proof.
"There exists a well-ordering on the powerset of the powerset of the powerset of the powerset of the reals, PPPPR. Proof: Axiom Of Choice."
wow, thx u

[Anonymous No.12230811]

>>12230779
Nobody suggested it was.
>>12230787
Delusional. It's still an open problem. If you're thinking of Cohen, he only showed that it's independent of ZFC.

[Anonymous No.12230826]

>>12230811
Then how do you think it can be solved?

[Anonymous No.12230830]

>>12230826
I've no idea lol, that's why it's still an open problem. If I knew, I would solve it myself.

[Anonymous No.12230836]

>>12230830
Does it have a solution?

[Anonymous No.12230842]

>>12230836
That person is on some kind of Platonism trip where they think the question is well-formed without reference to a particular axiomatic system.

[Anonymous No.12230843]

>>12230836
No, it's still an open problem.

[Anonymous No.12230851]

>>12230842
That's exactly why I'm trying to make him realise what he is saying.

>>12230843
I did not ask that. Does there exist a solution?

[Anonymous No.12230860]

>truth = provability in ZFC

[Anonymous No.12230868]

>>12230842
Up until 19th century people have been doing mathematics perfectly well without reference to a formal system. You're delusional.
>>12230851
>Does there exist a solution?
How would I know?

[Anonymous No.12230870]

>>12230868
There exists in the alien box we excavated, though.

[Anonymous No.12230883]

>>12230860
We're not saying that, but we are saying that the question doesn't even make sense outside of any formal system.

[Anonymous No.12230892]

>>12230883
Why not?
People have reasoned about mathematics well before formal systems came along.

[Anonymous No.12230899]

>>12230868
>doing mathematics perfectly well
I wouldn't call absolute bullshit proofs of basic things like the monotone covergenge theorem «doing mathematics perfectly well».

[Anonymous No.12230903]

>>12230899
That doesn't matter. What matters is that a very large portion of mathematics done before formal systems came along is good, sound mathematics.
This alone invalidates your claim that mathematical questions are meaningless without a reference to a formal system.

[Anonymous No.12230905]

>>12230892
Yes but when they did enough work they realized that their naive conception of mathematics contained contradictory statements. We use formal systems for the same reason we use logic to do math. The whole discipline quickly degenerates into meaninglessness if we don't.

[Anonymous No.12230910]

>>12230903
Your definition of meaning is much weaker than mine. I would say that things like happiness or humanity are also meaningless from a mathematical perspective.

[Anonymous No.12230911]

>>12230899
What was this BS proof?

[Anonymous No.12230918]

>>12230911
Look up Dedekind and the real numbers. Basically until the 1880s or so mathematicians literally just argued that if it seemed like numbers got pretty close then they did. Their conception of the real numbers was actually weaker than that possessed by an undergraduate taking a real analysis course today.

[Anonymous No.12230922]

>>12230910
>I would say that things like happiness or humanity are also meaningless from a mathematical perspective.
Happiness and humanity are not mathematical concepts.
>>12230905
None of what you said contradicts what I said.

[Anonymous No.12230925]

>this madlad actually believes he is not using a formal system just because he never refers to it
The absolute state of /mg/

[Anonymous No.12230927]

do you guys have any good recommendation for a probability theory book?

[Anonymous No.12230934]

>>12230925
Anon said "without reference to formal systems" not "without there existing a formal system under which the steps can be justified". You're a moron.

[Anonymous No.12230936]

>>12230922
Okay, I would also say the idea of a square is meaningless outside of a formal axiomatic system. Which isn't to say thay you must do all your work that way, but you'll realize after about five seconds of trying to study anything about «a square» that you'll have no idea what the fuck you're talking about without a bunch of axioms.

[Anonymous No.12230939]

>>12230883
I'd weaken that a bit by saying "formalizable system"

>>12230892
That's generally a fair case you make there, but one must also consider that pre-1930 math was a world where mathematicans didn't conceive that fully formal arithmetic would leave open some universally quantified statements. They didn't think mathematical questions, to be answered in a indistputible way, need a hard syntatic foundation.

And the issue is "solved" by Cohen relative to some framework and that's in a way the only agreeable answer you can get.
Otherwise, I can "solve" it right here:
There is a sequence of injections between infinite sets
[math] {\mathbb N} \to \{0,1\}^{\mathbb N} \to {\mathcal P}{\mathbb N} \to {\mathbb R} [/math]

But for a hand-on notion of "function", there is no injection
[math] {\mathbb R} \to \{0,1\}^{\mathbb N} [/math]

So [math] \{0,1\}^{\mathbb N} [/math] violates the continuum hypothesis and we're done.

The
>hand-on notion of "function"
is of course relative to your system of choice, just like the question people discussed above, asking whether function-extensionality holds.

The question
>Are two functions equal if all their return values are equal
can't be settled for all systems at once, and many of them can by consistent but have different answers.
CH is the same kind of question as the "function" extensionality question.
There's many good notions of functions, just like there are many notions of the reals.
If you fix a system, you can get an answer. For ZFC set theory, CH is answered in that sense.

[Anonymous No.12230944]

>>12230927
Depends on desired level but I liked Feller's probability books and Billingsley's Probability and Measure.

[Anonymous No.12230964]

>>12230944
I was looking for something in the early graduate level. Im already familiar with measure theory and lebesgue integration. I'm gonna check your recommendations, thank you very much

[Anonymous No.12230968]

>>12230939
>just like there are many notions of the reals
What do you mean by this?
What are the different notions?

[Anonymous No.12230976]

https://sites.google.com/view/purduetopologyseminar
>The Purdue Topology Seminar is held Wednesdays 11:30am - 12:30pm unless otherwise noted. We are currently meeting online through Zoom. Every Monday before the seminar a Zoom link will be shared in our email list.
>We have begun to record our online seminars. We publish them on our YouTube-Channel.
https://www.youtube.com/channel/UCLJWRcHW-mjDSbpkIX6B-cw/

PhD student & post doc positions in Norway
https://www.jobbnorge.no/en/available-jobs/job/194496/phd-fellowship-in-algebraic-topology-and-tensor-triangulated-geometry
https://www.jobbnorge.no/en/available-jobs/job/194493/postdoctoral-fellowship-in-algebraic-topology-and-tensor-triangulated-geometry

[Anonymous No.12230992]

based or meme?

[Anonymous No.12231014]

>>12230992
yes

[Anonymous No.12231026]

Pls help.
What is the name of the smallest non-interval graph?

[Anonymous No.12231030]

>>12230968
If you're speaking formally, then e.g. take ZF minus the axiom of infinity. The reals are a now merely a proper class. Do functions on classes exist? What's a class? Are the Cauchy reals and Dedekind reals isomorphic? What does it mean to be isomorphic? Does it have to do with functions?

[Anonymous No.12231035]

>>12230968
If you're speaking formally, then e.g. take ZF minus the axiom of infinity. The reals are a now merely a proper class. Do functions on classes exist? What's a function? Are the Cauchy reals and Dedekind reals isomorphic? What does it mean to be isomorphic? Does it have to do with functions? Is the order on the reals decidable? You can make a lot of choices there.

[Anonymous No.12231045]

>>12231014
both?

[Anonymous No.12231068]

>>12231026
I think that's actually just the square. But if you meant pic related, I don't think it has a well-known or standardized name, but I'd call it a 3-asteroid graph and say that an n-asteroid graph is an n-cycle with an extra edge hanging off each vertex. You should make sure to mention what definition you're using though and not expect people to be able to just look it up.

[Anonymous No.12231076]

>>12230968
I just mean different definitions of the reals in different axiom system

I wanted to cook up something that's not just the standard computer implementations, but I actually don't want to incentivise Tooker or Wildberger memery here

[Anonymous No.12231079]

>>12227806
Wouldn't A-(A U B) be empty
Is it supposed to be intersection?

[Anonymous No.12231082]

>>12227811
-_-

[Anonymous No.12231099]

>>12231068
my bad. it was a dumb question in any case. thank you.

[Anonymous No.12231113]

https://mathoverflow.net/questions/156238/function-extensionality-does-it-make-a-difference-why-would-one-keep-it-out-of

[Anonymous No.12231126]

Can you integrate an integral that isn't monotone? In Rieman? In Lebesgue?

[Anonymous No.12231129]

>want to practice my differentiation
>use cengage's practice tool
>get this
what the fuck? Why would they write the answers like this instead of [eqn]\frac{3-4v^3}{2\sqrt(3v-v^4)}[/eqn]

[Anonymous No.12231140]

I know that a Laurent series is a further generalization of a Taylor series, but is there a further generalization of Laurent series or is that the most general you can get?

[Anonymous No.12231143]

>want to practice my differentiation
>use cengage's practice tool
>these are the options for answers
What the fuck? Why would they write the answers like this instead of
[eqn]\frac{3-4v^3}{2\sqrt{3v-v^4}}[/eqn]

[Anonymous No.12231192]

>>12231126
Sure, sin(x) isn't monotone but it's definitely Riemann integrable.

[Anonymous No.12231210]

>>12231126
Yes.

[Anonymous No.12231270]

After the proof ends, I usually stand up and start walking around my apartment rapidly. I question myself: "was it Bourbaki?" I keep repeating the question for tens of minutes, until I come to a conclusion.
>"No!"
Then I get angry. I pick up the pencil, and start writting: "This is geometry, this is handwaving. This is geometry, this is handwaving." I do this until I exhaust myself after some hours. However, had the proof been Bourbaki? I would've answered
>"Yes, this was Bourbaki."
I'd then proceed to slow down my stride, and in a merry caligraphic style write down "This is Bourbaki, this is algebra. This is Bourbaki, this is algebra." I'd do that for a slightly shorter time than I'd do the geometry mantra. I'd go to sleep smiling, having witnessed yet another Bourbaki.

[Anonymous No.12231371]

If I'm considering the possibility of going to grad school, should I take a second semester of real analysis or a semester of complex analysis?

[Anonymous No.12231379]

>>12231371
You mean you're not required to take two semesters of RA and one of CA for your degree plan?

[Anonymous No.12231381]

>>12231270
what's your point, everybody does this

>>12231371
real analysis, complex analysis is just power series in disguise

[Anonymous No.12231393]

>>12231140
There is no furthest generalization of anything.

[Anonymous No.12231398]

>>12231270
And this, lads, is why mathematics is fundamentally about comparing dick sizes.

[Anonymous No.12231403]

>>12231379
No just a single semester in real for my b.s. Complex is technically a "graduate" class

[Anonymous No.12231414]

>>12231403
Interesting. I had to take CA for undergrad.

[Anonymous No.12231990]

I am so lonely....

[Anonymous No.12232001]

>>12224371
Only people who don’t belong in math don’t do this.

[Anonymous No.12232008]

>>12227412
Go back you fucking faggot. This board is so gay it hurts to even browse it anymore.

[Anonymous No.12232077]

>>12232008
>Go back
To?

[Anonymous No.12232249]

>>12227552
/mg/ is losing its edge :/

[Anonymous No.12232283]

How do you guys study math? It takes me forever to even read a paragraph because I feel the obsessive need to imagine various cases and make various minor proofs about any statement, to "paint the picture" so to speak. Kind of an issue.

[Anonymous No.12232335]

*blocks your path*

[Anonymous No.12232349]

>>12230992
great book. just make sure to get the second printing. there's an erreta for the second printing online as well.

[Anonymous No.12232377]

>>12224197
Boys, I'm so bad at math, anything algebra and above is just pure fucking gibberish, I'm good at most other things but maths i'm fucking abismal at. Am I just plain dumb or severely retarded?

[Anonymous No.12232394]

Is there a branch of mathematicians dedicated to analysing the implications of statements that have not yet been proven true, to see if they create any hard to find contradictions? Such could be useful in disproving important and difficult problems, or by having a chunk of knowledge ready should they ever be proved

[Anonymous No.12232442]

Can someone give an example of a differential equation in 3D with a nonsmooth solution?

[Anonymous No.12232454]

>>12232377
Algebra is gibberish even to most algebraists. Why not get into something more visual or practical, analysis, geometry, computational stuff, or differential equations for physics?

[Anonymous No.12232470]

>>12232454
Anything that involves letters and formulas is fucking impossible for me, i always forget how to do it and i feel so fucking inadequate

[Anonymous No.12232478]

>>12232470
The more i think about the more shitty i feel anons

[Anonymous No.12232488]

>>12232454
t. babby who got filtered by group theory.

[Anonymous No.12232503]

>>12232394
It's more domain specific. For example a lot of results in number theory assume the Riemann Hypothesis.

[Anonymous No.12232524]

>>12232488
>t. newfag

[Anonymous No.12232626]

>>12232394
>Is there a branch of mathematicians dedicated to analysing the implications of statements that have not yet been proven true, to see if they create any hard to find contradictions?
It's called "trying to disprove the statement"

[Anonymous No.12232633]

>>12232442
>differential
>nonsmooth
I guess some use of a modulus or something

[Anonymous No.12232935]

>>12232470
The symbols represent tangible things, usually

[Anonymous No.12232951]

>>12232442
The equation [math]\frac{ \partial}{\partial x} f(x, y, z) = 0[/math] literally has uncountable nonsmooth solutions.

[Anonymous No.12233047]

Im doing an independent study on Algebra
My professor sent me an email pointing out a few places where i fucked up some problems, but im 100% certain im right on this one
Cuz there cant be more than one possible J, i can just repeat the process on the J =/= K to get that J
= ker = K, contradiction
Or even easier, the problem itself asks you to prove that there isnt even a single possible J, of course there couldnt be a second
Am i just losing it?

[Anonymous No.12233062]

>>12233047
Here is a rewriting to make the introduction of J more explicit
but that just makes the proof look more solid

[Anonymous No.12233068]

I have yet to see a non shit take on the incompleteness theorems

[Anonymous No.12233095]

Are all composite numbers divisible evenly by 2?

I'm trying to work out a way to know for sure.

[Anonymous No.12233101]

>>12233095
Lol wut?
21 is composite but it's not divisible by 2

[Anonymous No.12233103]

>>12233095
my favorite number is 15

[Anonymous No.12233106]

>>12233068
>I have yet to see a non shit take on the incompleteness theorems
Just do math then, why do you want a "take" in the first place? It's a formal theorem.

[Anonymous No.12233110]

>>12233101
Ok, thanks.

I'll be honest I just didn't want to go through them all dividing by two.

I'm not a retard, you know.

[Anonymous No.12233124]

>>12233110
>I'm not a retard, you know.
the definition of a composite number, is a number with more than one prime factor
if you dont want 2 to be there, then just multiply by two other primes

[Anonymous No.12233125]

>>12233110
You don't need to go through all of them you literally just multiply two prime numbers that don't equal two
You are a retard you know

[Anonymous No.12233137]

>>12233106
That's what I meant I just saw that other thread on sci about it and it made me wish people would just shut up about it

[Anonymous No.12233150]

>>12233047
Yeah it seems you are correct. It doesn't matter "how many" J there are because you're assuming there is one, then showing there couldn't be.

[Anonymous No.12233157]

>>12231371
Are you interested in geometry at all? Then take complex analysis.

[Anonymous No.12233161]

>>12233047
You're professor is wrong
It's akin to showing someone a proof of the irrationality of root 2, and having them say "that just means the fraction wasn't reduced to begin with" when in reality you assumed it was reduced and proved it wasn't

[Anonymous No.12233169]

>>12233150
>>12233161
sick, thanks guys

[Anonymous No.12233184]

Is there a good way to check my answers in a probability class, without looking at the answer key? I'm usually right, but there's just no clear way to be sure.

[Anonymous No.12233211]

>>12233184
Run a simulation?

[Anonymous No.12233354]

>>12233211
How? I'm stupid, I don't know how to do it.

[Anonymous No.12233396]

>Forgot to put +C when solving an integral

[Anonymous No.12233660]

>>12230754
Logic has nothing to do with truth. Truth value is not truth. So it's fucking worthless.
Maths has nothing to do with reality.

[Anonymous No.12233668]

>>12233354
Learn some python and learn how to use its random number generation. Then just program in whatever scenario you're asked to consider and run it a few million times.

[Anonymous No.12233729]

>>12225445
don't fuck up

[Anonymous No.12233834]

>understand the concept of simplifying limits
>forget how to do the math involved
I dont wanna go back to algebra and precalc stuff

[Anonymous No.12233920]

>>12233660
>Maths has nothing to do with reality.

Computers are not part of your reality?

[Anonymous No.12233984]

>>12233920
I'm not taking any side in your guys battle, but the "has nothing to do with" of course literally never applies.

The taste of cinnamon milkshakes relate to how Turing did on his marathons, I'm certain you can come up with a correlation.

[Anonymous No.12234628]

>>12233660
There's a coherent argument for math having as much to do with reality insofar as it is used in physics. See Quine.

[Anonymous No.12234642]

>>12230336
what a great pic

[Anonymous No.12234650]

>>12233834
>he doesn't revisit those concepts every so often
ngmi.
If you can't do it, it means you need to practice it more.

[Anonymous No.12234734]

What's the difference between
[math]\omega [/math] and [math]\aleph_0[/math]

[Anonymous No.12234747]

>>12234734
There is no difference, they are the same set. Most of the time we use omega to refer to the set of natural numbers, and aleph_0 when we are thinking of omega as a cardinal.

[Anonymous No.12234793]

>>12234734
[math]\omega_0[/math] (or often also called just [math]\omega[/math]) denotes the first infinite ordinal, i.e. the ordinal number such that all finite number [math]n[/math] have [math]n<\omega_0[/math].
There's also [math]\omega_0+1[/math] and [math]\omega_0+2[/math] and then [math]\omega_0+\omega_0+7[/math] and so on.

[math]\aleph_0[/math] is the smallest infinite cardinality. It is the cardinality of all the above sets. E.g. [math]|\omega_0|=\aleph_0[/math] or also [math]|\omega_0^3+18|=\aleph_0[/math].

Ordinal numbers and order types are a concept independent of set theory.
If you use set theoretic foundations, you can define the class of hereditarily transitive countable sets. In a theory like ZF, those are models for countable ordinals.
One then (usually) goes on to define limit cardinals as (represented by) the limit ordinals.
Note that "there are more" ordinals, since e.g. [math]\omega_0[/math] and [math]\omega_0+1[/math] have the same cardinal size.
But there's also theories of cardinals that set up things differently than that.

One might be tempted to just throw out cardinals and talk in terms of limit ordinals. But then cantor already found that there are definable sets for which there seem to be no functions relating them to ordinals. E.g. the set
[math]\omega\to \{0,1\}[/math]
has undefined cardinal size in the ordinal-identification sense. It's essentially arbitrary big w.r.t. limit ordinals.

[Anonymous No.12234797]

>>12234734
[math] \omega_0 [/math] (or often also called just [math] \omega [/math]) denotes the first infinite ordinal, i.e. the ordinal number such that all finite number [math] n [/math] have [math] n<\omega_0 [/math].
There's also [math] \omega_0 + 1 [/math] and [math] \omega_0 + 2 [/math] and then [math] \omega_0 + \omega_0 + 7 [/math] and so on.

[math] \aleph_0 [/math] is the smallest infinite cardinality. It is the cardinality of all the above sets. E.g. [math] |\omega_0|=\aleph_0 [/math] or also [math] |\omega_0^3 + 18|=\aleph_0 [/math].

Ordinal numbers and order types are a concept independent of set theory.
If you use set theoretic foundations, you can define the class of hereditarily transitive countable sets. In a theory like ZF, those are models for countable ordinals.
One then (usually) goes on to define limit cardinals as (represented by) the limit ordinals.
Note that "there are more" ordinals, since e.g. [math] \omega_0 [/math] and [math] \omega_0 + 1 [/math] have the same cardinal size.
But there's also theories of cardinals that set up things differently than that.

One might be tempted to just throw out cardinals and talk in terms of limit ordinals. But then cantor already found that there are definable sets for which there seem to be no functions relating them to ordinals. E.g. the set
[math] \omega\to \{0,1\} [/math]
has undefined cardinal size in the ordinal-identification sense. It's essentially arbitrary big w.r.t. limit ordinals.

[Anonymous No.12234938]

In python, I'm looking for an alternative for
product(range(n), repeat=d).

Namely, does anybody know whether there's a variant of this that only gives combinations up to renamings?
I.e. it should give me [0,1,2,1] and [1,2,2,0] but it shouldn't give me [0,2,1,2]
since that one is mere a renaming (1<=>2) of the first one.

[Anonymous No.12235007]

>>12234747
This is not true in general, it depends on your encodings. While ordinals are usually encoded as transitive sets, cardinals are equivalence classes of equipotent ordinals, and as such require the axiom of choice to be defined that way. You can work around this requirement, but then choice-free cardinals behave weirdly.

[Anonymous No.12235037]

Consider the [math]\sigma[/math]-algebra [math]\Sigma_n = \sigma(\{\{m\}\mid m\le n\})[/math], where the universal set is [math]\mathbb{N}[/math].
How can I prove that [math]\Sigma_n[/math] is at most countable? Alternatively, how can I prove that [math]2\mathbb{N} \not \subseteq \Sigma_n[/math] for all [math]n[/math] (where [math]2\mathbb{N}[/math] is the set of even numbers)?

[Anonymous No.12235066]

Consider the [math]\sigma[/math]-algebra [math]\Sigma_n = \sigma(\{\{m\}\mid m\le n\})[/math], where the universal set is [math]\mathbb{N}[/math].
How can I prove that [math]\Sigma_n[/math] is at most countable? Alternatively, how can I prove that [math]2\mathbb{N} \notin\Sigma_n[/math] for all [math]n[/math] (where [math]2\mathbb{N}[/math] is the set of even numbers)?

[Anonymous No.12235114]

>>12235066
To prove it's finite, simply prove by induction that if a set in the sigma-algebra contains a number greater than n, then it contains all the numbers greater than n.

[Anonymous No.12235151]

Need some advice on studying a specific area of math: game theory. I just finished calculus and am in a discrete math course. What other pre-requisites do I need?

[Anonymous No.12235168]

>>12235151
You don't need literally anything for game theory. Econ bachelors study it on their third or fourth semester.

Unless you wanna prove things, then you should memorize some fixed point theorems and brush up on your optimization.

[Anonymous No.12235192]

>>12235168
You literally do.

[Anonymous No.12235275]

What are your favorite graphs and their drawings?
I like this drawing of Petersen aesthetically, but that's a normie pick.
Some complete tripartite graphs have nice drawings too.

[Anonymous No.12235287]

>>12235275
What's so special about this graph apart from the symmetry?

[Anonymous No.12235299]

>>12235287
look at it's long Wikipedia article

[Anonymous No.12235362]

>>12235287
Besides what you can read in >>12235299, it's super useful in many proofs, and has been used as a simple counterexample for many theses in graph theory.

[Anonymous No.12235375]

>>12235275

[Anonymous No.12235403]

Can somebody please help with the following problem?

Let [math]x_m \in R^{\omega} [/math] where [math] x_m = (\frac{1}{m},\frac{1}{m+1},\frac{1}{m+2}, \cdots) [/math]. Show that [math] x_m [/math] converges to the zero sequence [math] 0 = (0,0,0,\cdots) [/math] in the product topology but not the box topolgy.

[Anonymous No.12235410]

>>12235192
Please detail.

[Anonymous No.12235416]

>>12235403
Where [math] \mathbb{R}^{\omega} = \prod_j \mathbb{R}_j [/math].

[Anonymous No.12235439]

>>12235403
In the box topology consider the box
(-1, 1)x(-1/2^2, 1/2^2)x(-1/3^2, 1/3^2)x ....
In the product topology it's obvious.

[Anonymous No.12235708]

>>12235439
Thanks anon. That helps.

[Anonymous No.12235720]

tfw the professor assigns proofs as hw question that are literally just in the book that he assigned

super tempted to cheat here lads

[Anonymous No.12235741]

>>12235720
All that's important is that you know how to do it for the exam, anon.

[Anonymous No.12235754]

>>12235720
It's not cheating if you understand it.

[Anonymous No.12235787]

tried to remember things about taylor series and this is what I found in 2nd google link

first derivative should be (1-x)^-2 right?

[Anonymous No.12235817]

>>12235787
yes, but (1 - x)^2 = (x - 1)^2

[Anonymous No.12235864]

>>12235720
I had this happen recently. The same proof on the HW as the example in the book.
What the fuck is the point of this?

[Anonymous No.12235882]

>>12235787
Correct.
But also you may want to write your answer in English. I don't think that garbledeegook from google is able to be understood.

[Anonymous No.12235912]

>>12235787
Based Polebro.

[Anonymous No.12235921]

How would I prove that I can’t have an unbiased estimator of f(mu)=1 if |mu|<1, 0 otherwise for the random variable X following a normal distribution of variance 1 and unknown mean mu?

[Anonymous No.12236012]

Is there any quick ways of determining if an arbitrary finite group is cyclic or not?

[Anonymous No.12236305]

tfw no dihedral group gf

[Anonymous No.12236425]

homework question:

So I want to convert the following into CNF: P => (Q => P)
This is what I've done:
= -P v (-Q v P)
= -P v -Q v P
= -Q

is this correct? Why do all the online calculators simply return "true"?

[Anonymous No.12236454]

>>12236425
How did you get the last line? Did you think you could cancel out -P v P cause it's a tautology? Cause you can't

[Anonymous No.12236459]

>>12236305
This but Tits group.

[Anonymous No.12236469]

>>12236454
>Did you think you could cancel out -P v P cause it's a tautology
yeah... wtf? why can't I do that?

[Anonymous No.12236481]

>>12236425
Also
-P v -Q v P
Is already in conjunctive form cause it's a shit ton of or statements that just so happen to be anded with T.
Or if that doesn't make sense to you, you can know it's in cnf because if you used demorgans laws you would get an and statement nested within negation, therefore you can't use demorgans laws in the first place

[Anonymous No.12236485]

>>12236469
You can only do that if it's a tautology anded with something. Imagine if I said "either true statements are true or op isn't gay" if I canceled out the tautology I would get "op isn't gay" which clearly isn't true

[Anonymous No.12236494]

>>12236481
>>12236485
I actually get it now. thanks

[Anonymous No.12236507]

>>12236485
actually I THINK I can get rid of "P v -P" from "P v -P v -Q" and replace it with "T" (or "true")
so I get
P v -P v -Q
= T v -Q
= T

[Anonymous No.12236520]

>>12236507
Yes that's correct, but there's a difference between replacing it with an equivalent statement and just completely cancelling it out. Like technically you could convert any tautology to cnf by just writing true but that's not really helpful lol

[Anonymous No.12236523]

>>12236520
no yeah, I agree. I was doing resolutions before and maybe that's why I was in the mindset of removing clauses completely. you're right

[Anonymous No.12236527]

>Statistics isn't mat-

[Anonymous No.12236559]

>>12228584
Engineer waves his hands, says the magic word "Radon Transform" and moves on.

[Anonymous No.12236563]

>>12235007
Ordinals are always transitive well ordered sets, that is the definition. Whether a model thinks a set which is not well founded or transitive is an ordinal or not is a different question. The first infinite cardinal is omega whether choice is invoked or not. Even in models where choice fails explicitly (where you have to use Scott's trick to define cardinals, like in models of determinancy) omega is the same set as aleph_0, that is, omega is the first infinite cardinal. Choice is only needed to say for all cardinals to be alephs.

[Anonymous No.12237037]

imagine believing in numbers

[Anonymous No.12237058]

>>12237037
I don't have to imagine.

[Anonymous No.12237133]

Hello

[Anonymous No.12237138]

>>12237133
I literally have a masters in mathematics and the ONLY correct answer is 1. You can rewrite the equation multiple ways to clarify. In PEMDAS, "m" is before "d" for a reason, and that reason is the distributive property. You would never, NEVER divide first because it removes the distribution from the paranthetical component of the equation. As an example, solve for X and you could never initiate by dividing both sides by 4 because it is literally attached to the parenthesis.

This equation floats around on social media by social engineers who want idiotic people to argue over this until everyone throws their hands up in the air and gives up, or resigns themselves to uncertainty. This is essentially a common core poll, to get an idea of what percentage of the population are fucking morons, but worse... opinionated morons even though they know themwelves that they're not specialists amd were probably shit at math.

Shut your fucking mouth.

[Anonymous No.12237166]

>>12237138
Hey, bud, it's for you.

[Anonymous No.12237342]

If a family of subsets W is a subset of P(A) where A is a set and W is closed under differences and contains A, is it a [math]\sigma[/math]-algebra? I've proved that it is an algebra but proving it is a [math]\sigma[/math]-algebra is a quite a bit more tricky it seems.

[Anonymous No.12237381]

I just turkey basted batter acid and gasoline into my thoroughly scarred ass hole rectal nigger dick hole. I can’t feel anything due to the scar tissue from nigger dicks and their rampant STDs. Will this combination of chemicals explode, will they explode if a throw a flaming cigar chaser in there?

[Anonymous No.12237401]

>>12237381
reddit moment

[Anonymous No.12237454]

>>12237381
Reddit moment

[Anonymous No.12237548]

>>12237381
reddit Moment

[Anonymous No.12237573]

>>12236563
>Ordinals are always transitive well ordered sets, that is the definition.
Nope, that's only one possible encoding, which is by far the most commonly used indeed. But just as Kuratowski pairs are not the only way to encode ordered pairs in ZF, there are many other encodings for ordinals. The only thing that matters is that they are isomorphic to equivalence classes of well-orders. FWIW with choice you could pick an arbitrary representative for every such class and call them "the ordinals".

[Anonymous No.12237579]

https://www.goodreads.com/book/show/186124.Fortune_s_Formula

Is this real? Has science gone too far?!?!?11?!!?2?!

[Anonymous No.12237600]

Would someone mind taking a look at these and telling me where I'm wrong?

C1 and C2 refer to C1 and C2 functions, respectively.

[Anonymous No.12237603]

>>12237600
[math]\mathscr{C^1}[/math] and [math]\mathscr{C^2}[/math] refer to C1 and C2 functions*

[Anonymous No.12237626]

https://youtu.be/3pRR8OK4UfE

So basically with this we can make something look more English like without knowing English, or any language. Pretty cool stuff...

[Anonymous No.12237655]

https://www.cs.virginia.edu/~evans/thorp.pdf

Let’s learn to count cards /mg/.

[Anonymous No.12237956]

>>12236563
By that logic, Gentzen's proof of the consistency of PA assuming ordinals up to \epsilon_0 would be one invoking a set theory involving infinite sets

[Anonymous No.12238015]

>>12237956
>Gentzen's proof of the consistency of PA assuming ordinals up to \epsilon_0 would be one invoking a set theory involving infinite sets
And you would be correct.

[Anonymous No.12238670]

what do we mean by "infinitesmally small" in differentials?
>inb4 limits
anything else?

[Anonymous No.12238686]

>>12238015
No, ordinals are not a set theoretical concept - just like rings are not a set theoretical concept.

[Anonymous No.12238708]

>>12238670
>"infinitesmally small"
>>12238670
>"infinitesmally small"
50/50 that it means [math]\lim = 0[/math] or "lesser than any positive number" which is again the same as being equal to 0. another 50/50 it's just some brainlet explanation which doesn't have any precise mathematical meaning

[Anonymous No.12238736]

>>12238670
infinitesmials are an outdated concept. see, epsilon-delta, achemedian principle, etc.

[Anonymous No.12238737]

>>12238015
I wouldn't call ordinal a set theoretical notion, but you can of course define your terms to have the word "ordinal" reserved.
One needs, in any case, not write down set theory in a way that the natural numbers are primarily modeled as a transitive set.

In any case, you can encode order types in theories of arithmetics without ever invoking [math] \in [/math] or rules about it.
And, to the contrary, I don't think Genzen's proof (involving [math]\epsilon_0[/math]) requires any set theoretical axiom.
Even if one would want to, it seems that that theory couldn't have Separation and Extensionality at the same time, otherwise you can model PA and your theory gets too strong.

[Anonymous No.12238741]

>>12238739
new
>>12238739