Advanced Mathematics Questions/Answer Thread: No elementary Calculus

>>7947048
Can someone describe what a type constructor and a bind/shove operator is and how it is used to define a monad?

>>7947048
>Calculus I-III
>advanced

Got a couple quick group theory questions

If H is a subgroup of a group G, then if a is an element of H, is it true that the right coset Ha = H?

Also, not sure, is the identity
element of a quotient group G/H just H?

>>7947055
Those are not allowed, anon.
Reading comprehension.

>>7947060
Yes and yes.
For the first equality, just note that [math]x \mapsto xa[/math] is a permutation of H (it follows from the fact that H is stable by multiplication)

>>7947048
Just hoping I could get a second opinion on a proof:

>Every finite set is compact

Let [math] \mathcal { A } [/math] be a finite set and let [math] \left \{ \mathcal { O } _{ \lambda } : \lambda \in \Lambda \right \} [/math] be an open cover for [math] \mathcal { A } [/math]. Label each element of [math] \mathcal { A } [/math] with [math] i \in \mathbb { N } [/math] pick[math] \mathcal { O } _{ i } \subset \mathcal { O } _{ i }[/math] such that each [math] a _{ i } \in \mathcal { O } _{ i } [/math]. Since [math] \mathcal { A } [/math] is finite [math] i [/math] is finite, thus [math] \mathcal { O } _i [/math] is a finite subcover for [math] \mathcal { A } [/math], which means [math] \mathcal { A } [/math] is compact.

Suppose G is a group, G_{i} are subgroups for i \in I and H is a subset.

How can we simplify \bigcap_{i \in I} G_{i} H?

Calc I-II I can understand, but I'm pretty sure most of the topics in Calc III are more complex than
>>7947060

>>7947150
They're not because he is clearly a math major if he is taking a course that requires him to have this knowledge, what op meant to say was
>math thread, no engg fags allowed

>>7947147
It's all right except where you say 'i is finite' instead of 'the subcover consisting of O_i is finite'

>>7947147
concise and simple, right idea and right execution. there's a typo at [math]O_i \subset O_i[/math] (your notation isn't helping you, give a name to the cover) but otherwise it's perfect

>>7947148
use math tags and ill answer

>Calculus III
>advanced math

This is what /sci/ actually believes.

>>7947169
>I can't read an I'm stupid
>I'm also a faggot

we know

>>7947159
>They're not
How the fuck's group theory harder than multiple integrals in different coordinate systems, or surface integrals?
>he is clearly a math major if he is taking a course that requires him to have this knowledge
Or a compsci with a math minor, and I KNOW how much you guys love them!
>>7947169
>>7947170
He was talking about my post, but I didn't say it was "advanced," just that I don't see how elementary math is less elementary on account of being discrete, rather than continuous.

>>7947161
>>7947162
Thanks guys.

>>7947147
Write [math]\mathscr O := \{\mathscr O_{\lambda} : \lambda \in \Lambda\}[/math] to help you specify subcovers more clearly. I wouldn't use script font for a topological space, but this is really nitpicky.

>>7947048
How do I solve
[math]\lim_{x \rightarrow 0} \frac{\sin x}{x}[/math]?!?!?! xddd

>>7947176
I meant to use mathcal like you rather than mathscr, but whatever.

>>7947179
Cancel out the x's

Sin0 = 1

btw Im an engineer

>>7947174
The point is the guy's doing math and not engineering watered down shit. Group theory is 100% no disagreement allowed harder than anything in calculus.

If you wanna talk about integrals, ask a question related to analysis, not retarded trivial computations.

>>7947182
dont feed the trolls, please. this thread is going decently ok, replying to shitposts doesn't help

>>7947061
Should have an or instead of a comma then. Learn grammar.

>>7947192
I'm not the OP, but the intent behind the sentence was clear, in my opinion.

>>7947196
Then you can't read. OP's sentence is bad syntax, and he needs to take a break from math to brush up on English.

>>7947179
>Squeeze theorem
>L'hopitals rule
>Expand sin(x) as a Taylor series, divide by x, evaluate at 0.

Take your pick.

>>7947183
>Group theory is 100% no disagreement allowed
As opposed to Calculus III, where there's more than one right answer to any problem?
>harder than anything in calculus.
You know, the more I come on here, the more I wonder if the only thing that you people call "trivial" undergrad math is the math that you're not able to easily consult google about.
"The infinite beauty of the triple set."

>>7947192
>i cant read
>i cant accept being wrong
what more do you want to add? stop littering the thread with your incessant idiocy

>>7947206
no disagreement allowed was a qualifier to "harder"

other than that i dont know what youre going on about. calculus is literally follow the instructions done to death in lecture and dont mess up the basic arithmetic

>>7947206
I think the point is that the thread is for discussing pure mathematics, and so discussion on the intro calculus courses (which are not particularly foundational for a mathematician) is unwanted. It is not a statement on the difficulty of any subjects.

>>7947212
>no disagreement allowed was a qualifier to "harder"
Right, and I'm wondering where you think there is any "disagreement" allowed in Calc III.
>other than that i dont know what youre going on about. calculus is literally follow the instructions done to death in lecture and dont mess up the basic arithmetic
I wouldn't even say that about Calc I or II, if you have a decent professor.
In Calc III, that definitely doesn't fly. It's like a physics course. You don't get some general formula that you can apply ad nauseam to solve problems. You have to actually understand what the theorem is saying and what you can do with it.

>>7947217
I guess that makes sense.

>>7947205

He's correct, but I'd add

>Squeeze theorem for calc I
>L'hopitals rule for calc I / II
>Expand sin(x) as a Taylor series, divide by x, evaluate at 0 for calc II

>>7947206
Dude. You're in engineering. Please leave I even specified what op meant. Go create your own thread titled calc 3 I harder than group theory.

>>7947237
Have you seen what most questions in a calc 3 tests are like?
>find domain of function this function of 2 variables
>calculate the fernet apparatus of this function
>calculate the line integral of this
>calculate the surface integral of this
That's their final exam.

We are talking about calc 3. Not analysis, it is literally follow the formulas and remember a couple tricks about integrals and you're golden.

>>7947179
L'Hopital's rule. This is calc I.

>>7947324
L'Hôpital**

OP said no baby math so I posted precalc questions

>>7947054
I'm not a programmer, so I don't do type theory stuff, but I know some category theory, so I can give you examples of monads and algebras of an endofunctor and tell you how they work.

>>7947203
Your comma is probably bad syntax.

>>7947179
sinx/x = (sinx - sin0)/(x-0).
So lim(thatshit)= sin'(0)= cos0= 1

I have a group of the cartesian product of the integers, what operation would this group be under.

>>7947390
+

>>7947390
+ since you can have inverses, if you had x, you wouldnt have inverses and division by 0 would go kaput

>>7947055
>>7947169

I have made the important words capitals just so you understand.

>No Pre-Calculus, Calculus I-III questions ALLOWED.

It was one sentence, even with the introduction of a comma to separate Pre-Calculus it is clear he meant no Calculus I-III
are allowed.

Maybe you egg heads should try reading words.

>>7947048
Are fourier transforms calculus

>>7947426
Nah

>>7947451
Can you explain them

>>7947455
Describing a periodically repeating wave as a mathematical function.

To compute them you first need to calculate the coefficients then apply them to the formula.

Are Lapace transforms allowed?

>>7947743
No thats difyq

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>>7947352
By type constructor he thinks of the endofunctors action on objects and operations like bind are concatenations of what you get from the functors action on arrows and the nats.

E.g. in Haskell where inhabitance is written with an infix ::,
for a monad 'm', an object 'a' and a term x :: m a
(a global element of the object m a)
if 'fmap' is the action of 'm' on arrows, and where they call the unit and co-unit of the monad
return :: a -> m a
resp.
join :: m (m a) -> m a

you can define the 'bind' operation

bind :: m a -> (a -> m b) -> m b
by
(x bind f) = (join . (fmap f)) x

The do this because the more complicated bind operation is what they usually need in programs, and join can be defined in term of it:
join y = (y bind id)

That is, they avoid thinking about what the co-unit of the monad is and so stuff looks complicated.
I've written down two basic examples of use (of return and join) in pic related

>>7947783
how the hell are fourier transforms okay then?

>>7947831
underage b& plz, also this is calc 1 material. So no, you didn't read correctly.

>>7947843
says Calc I to III is allowed. precalc was my stuff from last year.

>>7947845
>no precalc, calc I, II, or III questions allowed
Fixed OP's intentions for you.

>>7947831
Mods plz

Let V be a representation of some group G.

Is the "diagonal" i.e. [math] \{ v \otimes v | v \in V \} [/math] a direct summand of the tensor power [math] V \otimes V [/math]. I think yes, but I am not quite sure, since my teacher was not sure either.

Since G acts by automorphisms, i.e. bijective on V the diagonal should be a subrepresentation and the complement should be aswell.

Can anyone confirm/counterexample?

>>7947845
No, you misread. Advanced Mathematics Questions/Answer Thread: No elementary Calculus questions allowed

Yours is a trivial elementary question that doesn't belong in this thread. GTFO and read.

>>7947390
Just for completeness, you did not specify an operation, but yes the canonical, i.e. pointwise group action with + works. However it could also be the semi direct product, if one groups also acts on the other by automorphisms. Since interger addition is commutative it doesn't fucking matter though in this case.

Looking back I just realise this is a terrible post and will probably just confuse you more. So just disregard that, I suck cocks.

>>7947862
This is only true if you take the tensor product of a representation with itself. For the tensor product of two different representations, the diagnoal will not be a subrepresentation.

>>7947884
Yeah, sure. Thats why I said tensor power [math]V \otimes V [/math] and not tensor product like [math] V \otimes W[/math]. In the second case talking about "the diagonal" doesn't even make sense.

Thanks though, was just looking for quick affirmation to see if I am retarded.

>>7947801
Excellent answer, but for someone that isn't a programmer can you explain what ::, and Ma is & what all of that put together (e.g-- bind :: m a -> (a -> m b) -> m b) means in english / an example?

the notation makes no sense to me. I tried to watch a video on monads and understood up to definition of monoid but they didnt explain Ma and bind operator well

>>7947165
autism detected
>>7947148
It's just the (intersection of Gi's)H

What's the fastest way to factor a ring over a finite field?

>>7947055
Your fucking ten ply, bud

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>>7947324
>>7947339
>>7947205
itt people who don't know math.

good luck using lhospital, taylor series or anything involving a derivative you retards

computing the derivative of sin involves computing the very limit you want to prove things about, so it's a no-go

one correct answer I have seen is an involved, but easy to understand proof using the geometry of the unit circle.


every time I come to this board it becomes more clear that no one here knows what theyre talking about. Yes this is calculus 1, but you've demonstrated that you can't even do that properly. Ignorant, pretentious assholes

Show that the wf [math] \forall x \forall y \exists z (z \neq x \land z\neq y \land A(z)) [/math] is true in a normal model of M of a theory with equality if and only if there exist at least three things in the domain of M that have property A.

I know this has something to do with how normal models are defined using equivalence classes, which means that [math]z[\math] can't be the same object under each [math] \forall [/math]. My model theory isn't where it needs to be.

>>7950197
I'm retarded

>>7948825
You're wrong. El hospital works well.

>>7950287
By differentiating sin at 0, you are implicitly evaluating the limit you're looking for.

>>7948825
why use the squeeze theorem when the hospital has an ez fix

FUCKING ENGINEERS SHITTING UP THIS BOARD WITH THEIR OWN INEPTITUDE THERE ARE DOZENS OF RESOURCES OUT THERE FOR YOU TO LEARN FUCKING CALCULUS YOU STUPID SMUG PRICKS JUST GO GOOGLE YOUR DAMN QUESTION

>>7950358
hjälp me solve [math]\sum_{n=1}^1 n[/math] pls

>>7950354
Squeeze theorem is more elementary, you don't need to know anything to prove it. I would like to see some of the people here give a proper statement and proof of l'hospital's rule (it is not complicated but there is stuff to say)

>>7947060
H is a subgroup hence is a group itself so for any a in H operated with an arbitrary element in H gives you an element in H.

Yes, the identity element of G/H is H. I can't really comprehend how you would not know this one but hey, there you go.

>>7950362
I love the elegance of the proof of L'Hoptal's rule once you get analaticity out of the way. Literally just apply the definition of the derivative.

>>7950368
You're just not into math. He's the one who's making sense here from a mathematical point of view.

>>7948825
>taylor series or anything involving a derivative
The series is the definition of sine. That it agrees with the taylor series is a mere consequence of the definition. Refer to the "special functions" chapter of your analysis book. Though I agree, l'hospital is right out.

>>7950368
I'm baffled about how you came up with that from what he said.

>>7950368
>"U CAN NO NUFFIN" philfag
It's literally a circular argument

>>7950368
That's not what I was saying, you mong. What I meant is that the squeeze theorem basically follows from the definition of a limit.
L'hospital's rules has differentiability assumptions and writing a proper statement and proof requires you to distinguish cases (if you are trying to find the limit of f/g, you have to distinguish each possible indeterminate form and do a proof in each case, which becomes interesting when f'/g' is also an indeterminate form).
A good way to handle this sort of things is with asymptotic notation, which is both more efficient and more flexible. Besides once you know it, you never need l'hospital's rule again (which is why they don't teach it where I live)

>>7950390
How I've seen these series expansions is from proving series expansions are unique and then using the Taylor series to show there is a series expansion and then that is it's series expansion since it is unique.

I've never seen sine as literally defined as it's series expansion, I've always seen it as something that needed to be proven as well as the fact it is differentiable everywhere. And to start proving that, you need to know what the derivative of sine is and hence the derivative of cosine as well all of the nice derivative properties.

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>>7947182

>>7950361
You're starting at one and ending at one. You're not adding anything, so the answer is 0.

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Hey /sci/, could you please verify pic related? I tried using Latex in the /sqt/ but it screwed me over.

>>7950439
nah its 1.

you start at 1, so n=1. then pass is finished...

>>7950460
drats, you caught me

>>7950390
Wtf is the motivation of having some random ass series expansion pulled out of thin air

It only makes sense to define sine geometrically

You are literally too fargone how about you go find your way back to/pol/, sound good to you? Okay bye

>>7950454
I can't follow. What statement are you trying to prove? You should be using complete sentences for each step.

>>7950485
that [math]\mathbb{R}^* [/math] is isomorphic to [math]\mathbb{R}^* \times \mathbb{R}^* [/math], and hence of the same cardinality. I'm making use of the isomorphism theorem.

I'm tired and latex is giving me a hard fucking time

>>7950499
[math]K \subset \ker (det)[/math] but the other direction is not true. For example, consider [eqn] A = \begin{pmatrix} 1/2 & 1 \\ 0 & 2 \end{pmatrix}. [/eqn] We can see that [math]A[/math] is not an element of [math]K [/math], but it is in the kernel of det as it has determinant 1.

>>7950509
Oh I see, thanks
I was sure there was something wrong in there

Let [math]M[/math] be a Kähler manifold.

And let [math] {T_h}^*M [/math] denote it holomorphic tangent bundle.

How can I prove that [math] {c_1}\left( {{ \bigwedge ^n}{T_h}^*M} \right) = \frac{1}{{2\pi }}\left[ \mathcal{R} \right] [/math]

(where [math]\left[ \mathcal{R} \right] \in {H^1}\left( M \right)[/math] is the cohomology class of the Ricci form.)

>>7950629
>tangent
cotangent*

>>7950629

too gradschool for me bro, but it's sexy looking

>>7947179

test values close to 0 to see that the function appears to be approaching 1.

then do an epsilon delta proof using the definition of limit of a function

Suppose f is an analytic function f that has maximal domain in the sense that for every x for which there exists an analytic extension of f with x in its domain, then x is in the domain of f.

Suppose a particular Taylor series for f, centered at a, has finite radius, and converges to z at some point x on the boundary. Then is f(x) = z?

True for all points within the radius of convergence, but is it true at the radius of convergence of the series converges?

Can you please solve the Schrodinger Equation associated with the Helium Atom?

>>7947048
question about abel ruffini theorem: wikipedia says there's no general solution for all roots of polynomials degree 5 or higher. Are all polynomial roots still computable? Can all roots still be expressed in terms of radicals?

>>7951012
The fundamental theorem of algebra states that the roots of a polynomial will always exist; the Abel-Ruffini theorem states that there is no general way to compute the roots of polynomials with degree 5 or greater using addition, subtraction, multiplication, division, and exponentiation by rational numbers. For example, there is absolutely no way to write the solutions for [math]x^5-x-1=0.[/math] There are some are that computable, such as [math]x^5 =1[/math], but really saying that there is no general solution for roots of a polynomial is the same as saying there the roots cannot be computed(all in terms of radicals).

>>7951012
As for the first question, we can certainly approximate roots to arbitrary precision. I'm not exactly sure what you mean by computable though.

As for the second question, wiki also says
> That is, the theorem does not assert that some higher-degree polynomial equations have roots which cannot be expressed in terms of radicals. While this is now known to be true, it is a stronger claim, which was only proven a few years later by Galois.

>>7950991
Mmmmm pretty sure it's a perturbative calculation right?

>>7951068
Yeah I got that; my question was more like are there polynomials with roots that must exist, but there's no convergent sequence with limit as that root. Maybe my question wasn't as closely related to the Abel Ruffini theorem as I thought though.

>>7951069
That's essentially what I meant in terms of computable. Thanks for the spoonfeeding too, I'll have to look into these polynomials without radical roots.

You don't understand the difference between a solution and the algorithm used to find it.

>>7951083
wat is the difference

>>7950454
The kernel of your first morphism is not K actually, it is the set [math]\begin{pmatrix} a & c \\ 0 & a^{-1} \end{pmatrix}[/math].
But deep down you can guess why this would not work right ? These are continuous morphisms so they induce homeomorphisms between the quotients. If you actually did prove this, you would have proved that R^* (that has 2 connected components) is homeomorphic to R^* x R^* (that has 4 connected components), which, besides the connected component thing, is absurd essentially because R^* is "one-dimensional" whereas (R^*)^2 is "two-dimensional".
The statement itself, is not true either because R^* has one element of order two : -1, whereas R^* x R^* has three: (-1,1), (1,-1), (-1,-1).
Interestingly though, you can prove that [math]\mathbb R_+^* \equiv \mathbb R_+^* \times \mathbb R_+^*[/math]. Again, because of the dimension problem, you cannot really construct the isomorphism but you can prove it using cardinality considerations (however I think you were trying to prove that they have the same cardinality by finding an isomorphism so that wouldn't work).

>>7951149
What really made me question me was my conclusion that R was isomorphic to R^2
Unfortunately for me I was too lazy to check the order of the elements which you pointed out

What is the maximum number of vertices in a regular graph of degree 3 with diameter 2?

>>7951352
Exercise: Prove that [math]\mathbb{R}^n [/math] and [math]\mathbb{R}^m [/math] are not homeomorphic for [math] n \neq m [/math].

>>7951352
Well funny enough, R is indeed isomorphic to R^2 as a group

>>7951857
Also, as a multiplicative group, [math] \mathbb{C}^* [/math] is isomorphic to the multiplicative group on the complex unit circle.

Im reading this paper about renormalized solutions of PDEs (not necessary to know what that is to answer the question) and stumbled upon the following statement (without explanation):
Let [math]u \in W_0^{1,1} (\Omega) [/math] that is [math] u [/math] is weakly differentiable on an open, boundend set [math]\Omega \subset \mathbb{R}^n [/math] then the set
[eqn] \{ x \in \Omega : | \nabla u(x)| > s, |u(x)| = 0 \} [/eqn]
has (Lebesgue-)measure zero for any [math]s>0[/math]. The gradient here is beeing the gradient in the sense of weak differentiation.
My idea for why that is true was the following: Suppose the set has not measure zero, then there exists an open, connected subset [math]U \subset \Omega[/math] on which [math]u(x)=0[/math] and [math]\nabla u(x) \not = 0[/math]. Since [math]u[/math] is even strongly differentiable (because it is constant) on [math]U[/math] and also weakly differentiable the gradient in the strong sense equals the gradient in the weak sense and is not zero, which is a contratidiction to [math]u[/math] being constant on [math]U[/math]. Does that work?

>>7951930
Not every set of positive lebesgue measure has a connected subset of positive measure. See the fat cantor set.

>>7951930
I've thought up a intuitive reason for the fact, though not a proper proof. If that set has positive measure then it feels like similar sets with the same definition but instead |u(x)|=epsilon should have positive measure for an uncountable number of epsilons near zero. Then, uncountable sets of positives sum to infinity.

>>7951959
Oh wow I did not know that one, thanks. Any ideas how to fix it?
>>7951980
Sorry I don't relly get your inution behind this, do you mind elaborate? The function is not continuous, only in the Sobolev Space [math] W_0^{1,1} (\Omega) [/math].

>>7947192
I agree with this even though I knew what he meant from the comma for some reason. weird

>>7951993
The intuition was that the weak gradient being bounded below on the set where u=0 forces the weak gradient to be bounded below on nearby |u| since weak gradients imply "upward movement" of some sort. Now that I think about it, the uncountable sum thing I said is totally invalidated by the discontinuity of the functions involved, but the proof, with more work than typical for an explained step, might be rescued by working with the integral equations the definitions give.

>>7952010
>explained step
I meant unexplained step.

>>7951959
>See the fat cantor set.
In the US, it's called the Sir Topham set.

>>7952010
Well yeah thats kind of the direction I was trying to go in with my argument above. But you mentioning the integral identities gave me the following idea:
If we set [math]A:= \{ x \in \Omega : | \nabla u(x)| > s, |u(x)| = 0 \} [/math] the by definition of the weak deravative we get for all [math]\varphi \in C_c^{\infty} (\Omega)[/math] the equation
[eqn]0 = \int\limits_A \! u \varphi '= \int\limits_A \! \nabla u \varphi [/eqn]. By the fundamental lemma of varionational calculus we get [math] \nabla u = 0 [/math] almoast everywhere on [math] A [/math] which implies that [math] A [/math] is a subset of a set of measure zero and so also has measure zero.
What do you guys think about this?

>>7952064
That will do it. Darnit I've must have seen that equality used like 20 times in Evans Ch.2. I feel bad for not thinking of that.

>>7952064
Well I kind of screwed up the last part. But with the right contratiction or choosing s>=0 this will do, right?

>>7952081
Yeah that equation gives the result.

>>7952081
Yeah me too, at least I didn´t have to got to my supervisor for this, that would have been quite embarrassing. Anyway, thanks for your help, it´s always nice to see there are some people on /sci/ who do math other than algebra.

>>7952088
meant for >>7952079

Can someone address this:
>>7947054

>>7953714
does this help
https://wiki.haskell.org/All_About_Monads#Type_constructors

or more generally
https://wiki.haskell.org/Monad_tutorials_timeline

>>7953789
The problem I have is I don't know Haskell and every tutorial I read is written using Haskell. I am fairly familiar with Python.

I understand mathematics and know what monoids are but whenever these examples use haskell syntax such as "(>>=) :: m a -> (a -> m b) -> m b" I have no idea what m_a is or what the notion really means. If you can enlighten me that'd be great to know what type/thing 'a', 'b', 'ma' is/represents. I really haven't found anything that explains what 'ma' is/does in a clear non-haskell way.

Also, I noticed a lot of examples use 'maybe' but don't define what it is/does/. So it leaves me extremely confused.

>>7947048
What is life?

>>7954003
Here you go.

http://www.dictionary.com/browse/life

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>>7948367
The first I said in the line
>E.g. in Haskell where inhabitance is written with an infix ::,
e.g.
7 :: Int
's' :: Char

If m is a functor or monad, m a is the object image. E.g. if m is mapping types a to lists of element of that type)
then
e.g.
3 :: Int
[3,7,6]:: m Int

Int -> Int is the type of functions from Int to Int, e.g.
f :: Int -> Int
f(x):=x^2
can be considered as such a function

bind :: m a -> (a -> m b) -> m b
or
bind :: m a -> ((a -> m b) -> m b)
means bind is a function that takes an element of m a and maps it to a function of type
(a -> m b) -> m b

consider e.g. the function that maps a number to the correpsonding word
f :: Int->String
f(7):='seven'
(the function definition is not Haskell syntax)
Consider now the function that doubles the output
f :: Int -> List String
g(7):=['seven', 'seven']
The return function (natural transformation) for the list monad is obvious:
return a -> List a
return(x):=[x]
The fmap is also clear e.g.
fmap:: (a->b) -> (List a -> List b)
(fmap(h))([x,y,z]):=[h(x),h(y),h(z)]
The join function (natural transformation)
join :: List (List a) -> List a
is flatten, e.g.
join([['hi', 'anon'],['kid']]) = ['hi', 'anon','kid'])

Finally, the (automatically deducable) join function
bind :: List a -> ((a -> List b) -> m b)
is defined as
bind(list)(f) := join((fmap(f))(list))
e.g.
with the function that does f(7):=['seven', 'seven'], you get
bind([3,7,6])(f)
= join([f(3),f(7),f(6)])
= join([['three','three'],['seven','seven'],['six','six']])
= ['three','three'],['seven','seven'],['six','six']]

The point is that by setting up functor/monad structures in your programming structure, you can use existence theorems from algebra that provide you free code.

(The cateogries, namely ones of types, let you do type checking and inference, which was a/the motivation for Haskell
https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system
)

>>7953962
People write
ma :: m a
.e.g
[3,7] :: List Int
Here ma=[3,7] is an element

>>7954035
that one g should also be an f, or otherwise I should maybe have used more different function names

>>7954003
I see highschool is out.

>>7954035
The (original) motivation for Haskell was the standardization of lazily-evaluated pure functional languages - there are and were other languages already that used Hindley-Milner-based type systems. But a lot of the new work on the main compiler (GHC) is focused on adding fancy new features to the type system.

>>7954035
Thanks for such a thorough response. What are good resources to learn both haskell and the mathematics it utilizes?

>>7954070
This is a good guide just for learning Haskell: https://github.com/bitemyapp/learnhaskell

The resources it recommends are good quality and contain exercises.

I don't know about any main resources for haskell-related math, mainly I find them in blog posts or papers online.

Resources for type theory may be relevant, and possibly also category theory. The use of category theory in Haskell isn't as much as people sometimes think, its main use is in designing good and reusable interfaces for things (e.g. monads). The people in the lens irc channel (#haskell-lens on freenode) seem to know a lot about category theory, though, and I think some of the lens library uses abstractions from CT in its design.

>>7954089
also many libraries designed/maintained by edward kmett

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>>7954059
You're right I shouldn't have downplayed the lazy aspect, but I just don't care for it much. I've read pic related.

>>7954070
Donno, I've not found a source I really like. "Learn Yourself" and "Real World" Haskell are the two standard intros. For an algbera & Haskell intro, there is this guy, maybe it's worth it
http://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/

When am I going to use [math]y = mx+b[/math] in my life?

>>7951837
That proof is surprisingly very complicated.

>>7954132
grade 6 is a little early to wonder about these things, just trust us

>>7954139
When will I use math in life XD
art > math
money and fame > dumb numbers

>>7954137
Prove there exists a one-to-one function such that [math]f : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}[/math].

>>7954160
Pretty easy desu

>>7947048
>no pre calculus or calc1-3 allowed

So discrete mathematics is allowed? lol..

>>7954160
(x,y) |---> x

>>7954220
not injective

Is the heaviside step function, defined on the reals, differentiable almost everywhere?

>>7954160
The people saying this is easy are wrong. The intuition is you want to construct a function which goes through [math]\mathbb{Z} \times \mathbb{Z}[/math] "diagonally". That is, if you graph the points of [math]\mathbb{Z} \times \mathbb{Z}[/math], your function should go through it in progressive diagonal rows. Something like:
[math](1,1) \to 1[/math]
[math](2,1) \to 2[/math]
[math](1,2) \to 3[/math]
[math](3,1) \to 4[/math]
[math](2,2) \to 5[/math]
[math](1,3) \to 6[/math]
...
I'll let you think about this for a while, but there are a lot more hints I can give if you need help. It's a tough problem to construct the actual function, even if you understand what it should be intuitively.

>>7954160
rudin 2.12

>>7947048
What is Mochizuki really trying to say here?

how do u graph polynomials

>>7954293
Okay I'll bite. http://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials.aspx

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>>7954132
Tell me what you're going to do with your life first.

>>7954091
>>7954089
Thanks!

>>7954242

Isn't this basically Cantor's proof for countability of rationals?

>>7954469
watch anime and shitpost about on it /a/
whilst intentionally posting algebra I - II and calculus questions on advanced math threads then fap to waifu and cry myself to sleep

>>7947048
Calc is the hardest kind of math stfu

>>7954660
t. 12 year old

Let [math]X[/math] be a Banach space, let [math]Y\subset X[/math] be linear subspace such that, for any [math]x^*,x'^*\in X^*[/math] if [math]x^*\neq x'^*[/math], then [math]x^*_{|Y}\neq x'^*_{|Y}[/math]. What can we say about density of [math]Y[/math] in [math]X[/math]?

>>7947055
And here I am waiting on the Precalc thread someone can explain this cookbook bullshit.

>>7954733
what do you need help on
tell me

>>7954646
""""""""Cantor's"""""""" """"""""proof""""""""

>>7954070
>>7954091
The bitemyapp guy is writing a book now. I have the pre-release versions if someone wants it...

>>7954781
Pre-calc is a requirement for cell culture please don't make fun of me.
I know it's absurdly simple to most of you but I can't figure out how to complete the square of a trinomal equation to find the vertex. Literally everything else I'm fine with but the subtle complexity of this shit eludes me.
One example wound be: y=3x^2 +4x -3

>>7954852
>try to rewrite expression as something squared
>subtract some number to remove any error

>>7954850
GIB ME BLS

>>7950362
There's basically noting to prove when you already have the mean value theorem proven

>>7950472
To have geometric definition basing on unit circle you first have to show what lenght of a circle is, which involves integration and differentiation, and I think that's harder than series expansion

>>7954879
have fun, anons

https://mega.nz/#!C18ihCqL
key:
!cVUvUrAicn7ylpQg6FhRDlxW0FE_nQ9qLiyONnQGULo

>>7954908
oh shit nigger

>The wide recognition of the power of Eilenberg and Mac Lane's ideas was a bit slow to come. Years passed before the words category and functor "could be pronounced unapologetically in diverse mathematical company" (Freyd). Luckily, the Air Force Office of Scientific Research became an important sponsor of research and conferences on category theory - an unexpected benefit of Mac Lane and Eilenberg's wartime work. Only by the mid-1960s did the Air Force eventually figure out that category theory was too abstract to bring any tangible improvements to air combat and discontinued its support.

>>7955221
>Only by the mid-1960s did the Air Force eventually figure out that category theory was too abstract to bring any tangible improvements to air combat and discontinued its support.
cucking the army like grothendieck did

From residue theorem

[math]\oint_{\mathbb{C}}dz~ f(z)\csc \pi z = \int_{-\infty}^{\infty}dx~f(x)\csc \pi x = \sum_{n \in \mathbb{Z}} f(n)[/math]

given [math]f(z) \rightarrow 0[/math] as [math]z \rightarrow \pm i \infty[/math] and that [math]f(z) \neq 0 [/math] at [math] z \in \mathbb{Z}[/math]. What if I want to find an expression for [math]\sum_{n \in \mathbb{Z}_{\geq 0}}f(n)[/math] instead? What integrand should go into the contour integral?

>>7955221
Can you describe what a functor and category is?

I read functors defined as 'containers' of elements. So whenever you apply functions to those functors you are actually unwrapping the containers and applying it to the elements and then wrapping the elements back into the container. Is this correct? What is the "container"? Another function, a list?

Also, what is a category? Is it a morphism between objects? Can it be a composition of functions?

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>>7955797

watch eugenia talking about categories

https://www.youtube.com/user/TheCatsters/playlists
http://www-bcf.usc.edu/~lauda/teaching/cat.html


a category depends on what you want. the simplest and most sterile definition is to say that it is a monoid.

the best definition is to say that it is a big bag, wherein you have
-objects
-arrows between objects
=>the arrows are what matter in a category, instead of the objects/elements like in your usual sets
-you can compose arrows


ex: in the category GRP of groups, the objects are the groups for instance, and the arrows will be the group morphisms

it turns out that focusing on the arrows, instead of the groups, is just as relevant as studying the groups, in order to learn about group theory.

The best part is that with category theory, everything is simpler, more natural, and more constructive.

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>>7955818

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>>7955822
even wikipedia has a book on category theory
https://en.wikibooks.org/wiki/Haskell/Category_theory


Harold simmons is pedagogical (but he works classically for advanced subjects)
http://staff.cs.manchester.ac.uk/~hsimmons/

and nlab
https://ncatlab.org/nlab/show/category+theory
Now, a functor is just a map from a category to a(another) category. the functors will be arrows for a bigger category, and the objects of this bigger category will be the little initial categories.
THIS IS what matters in category theory, that consider functors as arrows. Then there are 2-cells, which means new arrows who take a functor to another functor.

for functors, so you must map a category to another category, which means
the objects of the of first category will become some objects of the second category
the arrows of the of first category will become some arrows of the second category, with some property of keeping trakc of domains and codomains of these arrows.

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>>7955839
here is well explained

https://jaortega.wordpress.com/2006/03/17/programmers-go-bananas/

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>>7955797
The way I like to think of it is a category is a collection of objects connected by arrows along with a way of composing arrows which yields a third arrow given two arrows which are head to tail. A good and easy example is to have the objects be the class of sets, the arrows be functions between the sets, and the arrow composition be the same as function composition. Many important categories are that but with restricted kinds of sets and maps. However it's possible to make categories with properties that cannot be realized with sets and mappings between them. In this simple framework a functor is a mapping from one mess of objects and arrows which respects the structure the objects, arrows, and arrow composition form together. In particular it does not matter whether you walk down some arrows and then hop across the functor, or if you hop accross the functor and then walk down the arrows the functor maps to, you end up in the same place.

>>7955848
>in the same place
or doing the same thing rather
You don't just end up at the same object, but the path you took was equivalent.
Maybe a better way of saying it is that the functor distributes over arrow composition.

>>7955797
>I read functors defined as 'containers' of elements.
I saw this in a Haskell book. How do you reconcile this with the actual definition of a functor?

>>7955858
No idea. I have to go now though but I'll be able to look at it tonight.

>>7954132
Even people on Wall Street use very advanced math. You don't have to become a beta male STEM loser

>>7955858
a Haskell functor (eg. the list functor) takes types (like Int) to other types (like [Int])
so it's a mapping from the category of types (where the arrows are usually functions between them) to itself

>>7947147
congrats, you just solved cancer

[math]\int \frac{2x+5}{x^2-8x-9} dx[/math]

This was on my test, I solved it using trigonometric substitution, but I'm unsure.

>>7955888
Do you not understand how easy it is to check your answer?

>>7955888
Differentiate the antiderivative you got whilst taking the test, if you're led back to the integrand, you did it correctly. Otherwise, you got it wrong.

>>7947054
A type constructor is a type level function, for example in the C# programming language you have the type of List which is parametrized over some type: List<int> is just applying the type int to the type level function List.

A monad is some type level function (List / Nullable / etc) which implements a interface. Unfortunately you can not represent the monad interface in c# because it does not allow for higher kinded types! (you can not have a type variable that is a type function)

>>7955888
Read the OP. No calculus questions allowed in this thread. You want the stupid questions thread.

>>7956245
what is a dee over dix?

>>7947048
Why is the Mandelbrot set cut off after values that leave a disk of radius 2 around the point? Explain it to me like I'm in complex analysis, because I am.

>>7954828
thisss fammmm

>>7954702
It's pretty damn dense.

>>7956269
Bcuz thats how its definition

>>7956498
Haha no.

>>7956269
I'm trying to figure this out. The proof probably involves the parallelogram law, which you may not be familiar with.

https://en.wikipedia.org/wiki/Parallelogram_law#The_parallelogram_law_in_inner_product_spaces

With that tool in mind, see if you can do it.

>>7956540
>>7956498
>>7956483
>>7956269
what is a dee over dix

>>7955264
just finished complex analysis course, but can't you just take the integral over the Re>0 line?

>>7956753
Lmao I'm a fucking idiot.

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>>7955888
polynomial/polynomial should make you think partial fractions

>>7957373
partial fractions are kawaii desu

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HELP

>>7957397
there is no hope, kill yourself

>>7957397
See following thread
>>7950511
>>7954300

That is calculus,
gtfo of this thread

but.. you can break down that fraction into
[math] \frac{dt}{(t^2+1)^n}[/math]

Then use substitution.

I'm a non math major and have taken all the pre-reqs for upper division undergrad math courses. I like graphs. What class should I choose for next semester?

>>7957406
wrong

>>7957408
>I like graphs.
Which one? Graph of a function, or graph theory.

>>7957411
yff... noticed the error
Integrals of type (for n greater than or equal 2)
[math]\frac{ Ax + B}{(x^2 +ax +b)^n}[/math]
can be reduced to that fraction, but n<2.

>>7957433
tfw I was about to fix your latex but then you already did it after I hit update

>>7957433
n would have to be an integer for that to work, buckaroo.

>>7954702
Doesn't x* != x'* always imply the restrictions are not equal

>>7957487
never mind im retarded

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itt: beauty

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>>7957936
>categorical nonsense
>beauty
Lmao

>>7947048
Assuming pic related, calculate your odds of heart disease in the next 10 years.

>>7957397
You're dead

>>7950629
By the splitting principle, when computing characteristic classes of bundles, one may pretend that a rank k vector bundle is a sum of k line bundles. By the properties of the chern class, c_1(L_1 + L_2) = c_1(L_1)c_1(L_2). Note that if E = \sum_j L_j, then det E = \prod_j L_j. This leads to the conclusion that the first chern class of the holomorphic tangent bundle T is the same as that of det T.

The final step is to note that the first chern class of a bundle is minus that of the dual bundle, applied to det T and det T^*.

I read the doghouse explanation for groups and it was really cool. Does anyone have something like that for fields

>>7957373
+C hahahahahahaha

but yeah partial fractions

>>7958658
XD LOW IQ BRAINLOT

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>>7957944
>Homu

>>7957944
I was worried for a moment that I would have to live with some bilinear functor named Homu in my vocabulary. As if these weren't enough:
>homomorphism
>homeomorphism
>homology
>Hom
Question to people who know respresentation theory. Do most of the instances of the word "characteristic" actually correspond to things in respresentation theory? Like is there a correspondence between characteristic functions of sets and characters of some function space which comes from fourier analysis?

>engineering student
>I only understand the basic calculus stuff of this post. The other things look in mandarin for me.

Should I kill myself?

>>7959118
The character of a representation is the trace of the representation.
https://en.wikipedia.org/wiki/Character_theory

>>7954160
Let [math]f:\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}[/math] be defined such that [math]f(a)=(x,y)[/math], where [math]x[/math] and [math]y[/math] are the products of the factorization of [math]a[/math] such that [math]a=2^xy[/math], where [math]y[/math] is odd.

The fundamental theorem of arithmetic states that factorizations of integers are unique, so the function is 1-1. Letting [math]x\in\mathbb{Z}\times\mathbb{Z}[/math], we see [math]f(2^xy)=(x,y)[/math], so the function is onto. Since the function is bijective, it has a bijective inverse, thus proving the statement that there exists a one-to-one function [math]g:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}[/math].

>>7959131
I knew that, but I'm not going to learn something ridiculous like characteristic subgroups correspond to characters am I?

>>7959153
Oops,
change [math]x\in\mathbb{Z}\times\mathbb{Z}[/math] to [math](x,y)\in\mathbb{Z}\times\mathbb{Z}[/math]

>>7959154
Yes you will.

3x +5 =y
y=5

How do i solve for x?

>>7959154
Mathematicians aren't so creative with names usually. Words like normal, regular, perfect, and characteristic are used all the time whenever an adjective is needed.

>>7959181
use algebra to substitute and rules of arithmetic to solve.

>>7959181
[eqn]
\begin{aligned}
3x + 5 &= y \quad (y = 5) \\
3x + 5 &= 5 \\
3x &= 0 \\
x &= 0 \\ \\
\text{If } y &\in \mathbb{C}...\\
3x + 5 &= y \quad (y = 5i) \\
3x + 5 &= \sqrt{-25} \\
3x &= -5 \sqrt{-25} \\
3x &= -25i \\
x &= -8. \bar{3} i \\ \\
\end{aligned}
[/eqn]

>>7947048
What is 1+1=?

>>7959298
5 is in the complex numbers, you fucking moron. 5i =/= 5 ever.

>>7959298
Probably dumbest post I've seen here.

>>7959298
What the fuck? This is a new kind of retarded.

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How much time did you guys spend doing your master thesis? I have spent over 700h already but I am almost done at least.

>>7959406
Probably a semester or 2

Gicen a tournament T of order n, how many kings must there be if there is no super king (ie a kinng with all wins)?

Not homework, can clarify if not well written. Profesors did not have an answer, said to check for results in Ramsey theory

>>7959183
In set theory I recently encountered the adjective "remarkable" used to describe certain things. Found it charming.

>>7959298
This is certainly bait. Wouldn't be surprised if samefag as >>7959181. Not a bad thing. It's very funny.

>>7947192
>Anything

>>>else


>is fair game.

>>7959369
>>7959379
>>7959642
>>7959374
oh shit i'm laffin

>>7959123
Abstraction is useful only when you are getting something from it. Mathematicians use abstraction to achieve generality, but the kind of generality which is useful to mathematicians is not as useful to engineers. Trying to get abstraction for it's own sake is a fool's game. I'm not saying that you don't need to know theory; I'm saying you should study the theory you need to know rather than the theory math students use. Study PDE or something instead.

can someone give me an intuitive answer/feeling for the significance of the root locus analysis of a control system?

>>7947048
Curious if I have the right answer for the best linear approximation of f
[math]
f(x,y,z) = 4x^2 \sin (y) + z^4 e^{xyz}
\\
\textbf{u} = (1, 0,1)^t
\\

Df(x,y,z)= [8x\sin(y) + yz^5e^{xyz} \space \space \space \space 4x^2 \cos (y) + xz^5e^{xyz} \space \space \space \space z^3e^{xyz} + xyz^4e^{xyz}]
\\ Df(\textbf{u}) =[0 \space \space \space 5 \space \space \space 1]
\\P_1(f,\textbf{u})((x,y,z)-\textbf{u}) = 5y + z -1

[/math]

how do u find standard deviation

Are differential equations calculus? They are taught in calculus classes, but I've never actually seen them being classified as calculus.

>>7962873
Yes. Anything involving limits in some form is calculus.

If you really started hair-splitting maybe you could say that PDEs are analysis and not calculus because there's too much theory behind them, but the differential equations course you take in sophomore year or so is cut-and-dry calculus.

>>7947048
Firstly, why do I recognise the pic?

Secondly, how do I plot functions of more than one variable on paper? To give a silly example, lets take f(x,y)=x^2 +y.

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I don't understand why the functor between the category of topological spaces w/ smooth maps and the category of commutative rings of smooth functions on topological spaces (to R) w/ ring homomorphisms mapped through pointwise addition and function composition is a CONTRAvariant functor.

Can someone help? Why is it contravariant?

>>7963039
Have you tried taking three spaces U, V, W and explicitly writing down the map? You can do that right here.

Also, "of" functions?

Really, I only reply to point out how annoying it is that you write "w/" while writing out "homomorphism" 3 times in full

>>7963039
Literally just write down what the functor does to maps and you'll see why.
Basically, it's because given a map [math]f: M \to N[/math], the only reasonable way to "turn it" into a map between [math]C(M)[/math] and [math]C(N)[/math] is the map [math]f^*: g \mapsto g \circ f[/math] that goes from [math]C(N)[/math] to [math]C(M)[/math] (you use f to pull back functions on N to functions on M).
The same "argument" tells you why you can expect all sorts of "duality" functors (the one that takes a vector space to its dual, an abelian group to its dual, a manifold to its algebra of smooth functions, an affine scheme to its ring of global sections) to be contravariant

>>7963039
Because a smooth function is a map from X to R, and a map from X to Y induces a map from functions on Y to functions on X, by sending some Y -> R to X -> Y -> R.
>>7962897
There's also an algebraic/categorical notion of a limit - basically the limit of a sequence of algebraic structures (or a more general graph of them).

>>7962907
>Firstly, why do I recognise the pic?

One possibility is I used the same photo as the original thread and you saw both threads.

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>>7963157
Fair enough, it looks like a weird mix between this guy and EdibleNapalm.

>>7947048
yo how do i prove le convergence of [math]\sum_{n=0}^{\infty} \frac{2n}{n!}[/math]

>>7963275
The Ratio Test should be enough

>>7963275

ratio test confirm'd

Let p(x) be a polynomial. Suppose that the remainder when dividing p(x) by x + 1 resp. x - 1 is -1 resp. 1. What is the remainder when dividing p(x) by x^2 - 1?

>>7947048
How do I calculate the probability of a 3 state Markov chain being a particular state after n time with any transition matrix?

>>7963275
[math] \sum_{n=0}^\infty \dfrac {2n} {n!} = 2 \left( \dfrac {0} {1} + \sum_{n=1}^\infty \dfrac {1} {(n-1)!} \right) = 2 \, \sum_{m=0}^\infty \dfrac {1} {m!} = 2\, e^1 [/math]

if I have some function f(x) and it's nth derivative is D(n)
then I take the derivative of D(n) with respect to n, what would be some applications of that

Why is it possible to get a closed form expression of a sequence for some recurrence relations using generating functions? Is it because it's a countable set or something? Why are some other relations impossible to get a closed form from?

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>>7963431
You'd have to pass to something like the fractional derivatives, as you need a sort of smoothness in n

e.g. as in

[math] \dfrac{ d^a } { dx^a } x^k = \dfrac { \Gamma (k +1) } { \Gamma (k-a+1) } x^{k-a} [/math]

>>7963455
I'm afraid this borders Kolmogorov complexity territory

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>>7963039
Because function composition goes from right to left, you homosexual.

OP, please continue this thread when this ones dies... I'll be doing some analysis next week.

Also, I need a memory jog.
Polar coordinates. Is it r>0 or r>=0? Or is it both depending on choice of convention? Hopefully this thread is well read enough to kick my ass in the right direction.

>>7950454
Was confused, thougth that you meant R* as the dual space to R

>>7963716
What angle would you use when r is 0?

Algebra of [math]\forall[/math] and [math]\exists[/math]
How to get good at them?
Like when they commute, when they don't, when one statement absorbs another, and the complements of statements

Looking for a quick guide

>>7963803
In general they never commute, though may in specific situations modulo certain axiom systems.

You have to use your brain.

>>7963803
Literally just translate the statements to english and see what they mean.
Try to be as dumb as possible (there is no trick and there is only one possible interpretation of any given statement)

>>7963431
idk off the top of my head you could use it to obtain a sort of derivative representation of log factorials.
[eqn]
\begin{align}
\log(n!) &= \sum_{k=1}^n \log(k) \\
&= \left[ \frac{d}{ds} \sum_{k=1}^n k^s \right]_{s=0} \\
&= \left[ \frac{d}{ds} \frac{d^s}{{dx}^s} \sum_{k=1}^n e^{k x} \right]_{x=0,s=0} \\
&= \left[ \frac{d}{ds} \frac{d^s}{{dx}^s} \frac{e^{n x}-1}{1-e^{-x}} \right]_{x=0,s=0}
\end{align}
[/eqn]

Notably this representation is continuous in n and you can actually use it to evaluate the derivate of [math]\log(n!)[/math] at 0.
[eqn]
\begin{align}
\left[ \frac{d}{dn} \log(n!) \right]_{n=0} &= \left[ \frac{d}{ds} \frac{d^s}{{dx}^s} \frac{-x}{e^{-x}-1} \right]_{x=0,s=0} \\
&= \left[ \frac{d}{ds} (-1)^s \operatorname{B} (s) \right]_{s=0} \\
&= -\left[ \frac{d}{ds} s \zeta(1-s) \right]_{s=0} \\
&= \lim_{s \to 0} \left( s \zeta'(1-s) - \zeta(1-s) \right)\\
&= -\gamma
\end{align}
[/eqn]

>>7963778
Literally any?

>>7963812
>>7963836
Argh.
There's just too many to hold in my head at one time
Buffer overflow

Test: desu
To be honest

>>7963883
Then it's not a parameterization

>>7963906
Nobody expects you to hold eight in your head at once, but you should be able to hold at least three, e.g. the formal definition of a limit.

If you can't, practice.

>>7963939
I'm at 5 with a whole bunch of goofy qualifiers
> wth am I doing

What about changing a nexists to a forall and then switching the remaining foralls to exists and vice versa?
I remember seeing that in some book

I've messed up proofs before by translating them into English. I'd rather just get some guidelines

>>7964004
Yes, [math] \lnot \exists x \lnot [/math] can be replaced with [math] \forall x [/math] (and visa versa), and [math] \lnot \forall x \lnot [/math] can be replaced with [math] \exists x [/math], and visa versa.

You should understand why.

As [math] \lnot \lnot [/math] is equivalent to the null character, by the above we can also do things like replace [math] \exists x \lnot [/math] with [math] \lnot \forall x [/math], and visa versa.

>>7963883
You found the problem.

>>7964052
I guess derision is what I get for helping an anon on the internet.

>>7947048
How do I find the midpoint of a line when I know the coordinates of each endpoint?

>>7964108
For each coordinate you take the arithmetic mean of corresponding endpoint coordinates i. e. if your endpoints are (a, b) and (x, y) then the midpoint has coordinates ( (a+x)/2, (b+y)/2)

File: maxresdefault.jpg (38KB, 1280x720px) Image search: [Google]

38KB, 1280x720px

>>7952053
>fat cantor : fat controller : Sir Topham Hat

lel

>>7954227
yes

>>7963803
>>7963906
I'll use a simple example. Let [math]G[/math] be a group.
For every [math]g \in G,[/math] there exists an element [math]h \in G[/math] such that [math]gh = e,[/math] where [math]e[/math] is the identity element. (every element in a group has an inverse)
The negation of that statement is,
There exists an element [math]g \in G[/math] such that for every [math]h \in G,[/math] we have that [math]gh \neq e,[/math] where [math]e[/math] is the identity element.

Think about why those two statements are negations.

>>7962907
for a function of two variables, do 3 axes, x, y and f(x,y) and do like normal
for a function of more, you can't

>>7964974
>for a function of two variables, do 3 axes, x, y and f(x,y) and do like normal
I try that, but it comes out looking utter shit, and often misleading.

>>7965074
usually it's quite hard to represent with anything that isn't a parabola or a sphere

File: question.png (36KB, 928x482px) Image search: [Google]

36KB, 928x482px

I have no idea how to do this.

>>7965165
It's asking you to multiply two numbers and then plug it into the definition of the derivative.

>>7965176
I should have said this my first post, but it's more part (1) i don't know how to do

>>7965187
try by induction on n

Do there exist three circles such that one is perpindicular to every other, angle between 2nd and 3rd is 30 degrees, between 3rd and 4th is 70 degrees, between 4th and and 2nd is 79 degrees, also at no point do three circles meet at the same time. If answer is no, then proof why. If answer is yes, then give algorith how to build the construsction.

>>7965202
>Do there exist four circles
correction

Need some help with real analysis.

Given the function x^2 on [1,b], we define a partition P_n as follows:

x_0 = 1, x_1 = b^(1/n), ..., x_n-1=b^((n-1)/n) and x_n = b^(n/n) = b.

Show that for every epsilon > 0, we can pick n so that U(f,P_n) - L(f,P-n) < epsilon.

Help plos

>>7965404

Um, I don't remember the definition of your U and L sums, but what I expect you need to do is explicitly calculate U - L and then choose an appropriate n based on your result.

>>7965598
pls tell me this is bait

>>7965598
yes
no

>>7965701
H
E
L
P

P
L
O
Z

how do I prove that if I add 1 and divide by 2 as often as possible I will always end up at the number 1

>>7966228
just pick a couple random numbers and show that you get 1. QED

>>7966228
U srs? Add 1 repeatedly until you reach a power of 2.

we need to show that for every E>0 there is a an N: n>N implies 1<=x_n<1+E

We see that x_(n-1)>1=>x_n>1 and also that x_n>x_(n+1) hence that it is bounded and a subset of the compact set [1,x_0] and therefore it has a limit point and the sequence converges. Let 1+K>1 be be the value it converges to. Since it is monotonically decreasing this is the lower bound of the sequence.

Choose E>0 and N:n>N 1+K<=x_n<1+K+E.

therefore 1+K<=x_(n+1)<1+K+E. But we have x_(n+1)<=(xn+1)/2 therefore x_(n+1)<=1+K/2+E/2. Picking E<K/4 gives us x_(n+1)<=1+K/2+K/4<1+K so 1+K is not a lower bound which is a contradiction.

Therefore by contradiction there is no such K.

It's late and I may be a bit spacked.
>>7966352
is for
>>7966228

I want to see if I have this right about clopen intervals of R
For a<b
[b,inf) is clopen in (-inf,a]U[b,inf) because both (-inf,a] and [b,inf) are closed
[b,inf) is not clopen in (-inf,inf) because [b,inf) is closed, but (-inf,b) is open?
[b,inf) is not clopen in [a,inf) because [b,inf) is closed, but [a,b) is neither open or closed

Correct me where I'm wrong

>>7947183
No. Elementary group and representation theory is not harder than complex analysis.

>>7966610
>group theory is harder than calculus. try analysis instead
>No! group theory is not harder than complex anlaysis!

here's your (You)

>>7947048
Help! I've never understood Yoneda lemma and it wasn't ever a problem until now.

>>7966873
Don't worry, it has no content

>>7947179
this is b8

I never learned math in english, and now i have a problem that i cant solve with my current knowledge and i have no idea how its called in english, could anyone help me and give me some pointers?

The problem i have is something like having a 300-400 sided die and the question is how many times would you need to roll the die to get every single possible outcome, and how would standard deviation impact this?

>>7947147
A compact set is a set that is closed and bounded. Obviously, every finite set is closed because the complement of a finite set is open (take any point in the complement, then, you can always produce an open ball of radius min{d(f_1,p),d(f_2,p),d(f_3,p)....d(f_n,p)}, where f_i are the elements of the finite set and p is the arbitrary point in the complement), and finite sets are compact because you can make an open ball that includes every point from the set (make an open ball centered at f_1 whose radius is the sum of distances from each of the pairs of points in the finite set).

>>7966873
If you know all of the maps out of (or into) an object, then you know the object up to unique isomorphism.

>>7967024
This is wrong. Compact sets need not be closed in non-Hausdorff spaces. As a matter of fact, finite sets aren't even closed in non-T1 spaces.

>>7966565
Bump

>>7967379
Do you know what a bump limit is?

>>7966565
>[b,inf) is clopen in (-inf,a]U[b,inf) because both (-inf,a] and [b,inf) are closed
Yes. Proper clopen sets are the connected components of a space.
>[b,inf) is not clopen in (-inf,inf) because [b,inf) is closed, but (-inf,b) is open?
The statements "[math][b,\infty)[/math is closed" and "[math](-\infty,b)[/math] is open" are equivalent statements. The set [math][b,\infty)[/math] is simply not open by the definition of open sets in [math]\mathbb R[/math] (try to fit a neighborhood around b). Also, if it were clopen then [math]\mathbb R[/math] would be disconnected.
>[b,inf) is not clopen in [a,inf) because [b,inf) is closed, but [a,b) is neither open or closed
No, [math][a,b)[/math] is definitely open. The set is not open for the same reason as before.

The definition of closed is "complement is open." I'm not sure what you think it is.

>>7966981
Surely it's 400 times for a 400 sided dice?

Think of it this way, you'd only need to roll a 1 sided dice once, a 2 sided dice twice and a three sided dice three times. Surely this generalises? As for the standard deviation, everything's fucked if you don't assume the dice is fair.

>>7967402
> The definition of closed is "complement is open." I'm not sure what you think it is.
That a set has its limit points?
A set S is closed, if for every sequence in S that converges to L, L is in S.

I'm having a hard time keeping neighborhoods in my head, limits are easier to hold.
> Proper clopen sets are the connected components of a space.
Haha. I'm in the connected space section of my book. That's the only reason I started bothering with clopen sets.

> [a,b) is open
Ahhh. I thought those types of intervals would be neither open nor closed.

>>7967666
>That a set has its limit points?
This is helpful in certain situations, but the other definition will do you a lot of good to keep in mind (they're equivalent).

>I'm having a hard time keeping neighborhoods in my head, limits are easier to hold.
You should probably not do that. Limits are defined in terms of neighborhoods anyways.

>Ahhh. I thought those types of intervals would be neither open nor closed.
You're in the subspace topology. If [math]Y \subset X[/math], then the subspace topology on [math]Y[/math] has as open sets all sets of the form [math]U \cap Y[/math] where [math]U[/math] is open in [math]X[/math]. Definitions are important.

Drawing attention to my question >>7950811

>>7947390
sums=products in the category of finite groups.