Advanced Mathematics Question/Answer thread.

>>7985665
lol the list of what is not allowed grows more every day

>Advanced Mathematics Question/Answer thread.
>No Calculus I-III, Differential Equations, Partial Differential Equations, Linear Algebra, Real Analysis, Complex Analysis, Abstract Algebra, Topology, or Knot Theory allowed

>>7985665
>elementary Linear Algebra
By this, do you mean matrix algebra? I think we're starting to cut a bit too much out here. IT may not be advanced, but it could lead to decent discussion.

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This is now an IUT thread.

>>7985686
We can relax the restrictions a little.

This thread is below me. I've mastered triple integrals.

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>>7985686

How good is the Sage mathematics package? Will it be enough to cope with undergrad numerical analysis/algebra/linear algebra/diffy Qs or would I need mathematica/matlab?

>>7985737
You probably don't even need anything that extensive.

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>>7985713
Can a prime-strip be horizontally coric?

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Do Mochizuki runes make sense to anyone itt?

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>>7985756
R^{bad}

>>7985756
Well I's hope Mochizuki does, although I guess it could just be an elaborate hoax.

What is a functor? If you use any abstract algebra please define any definitions you use. I'm trying to see what it is in terms of functional programming. It has been described as a "container" which can contain functions, elements etc. that can be applied to some operation and then the result of that is put back in the "container".

Trying to get a better grasp of this "container" and what this thing is. Looking for an easy to understand explanation.

>>7985759
As in bad reduction you fucking plebeian.

>>7985767
Beware that the word "functor" means completely different things in math and in programming

>>7985767
In your programming language, you hav this collection C of types (Chars, Ints, lists of chars, lists of lists of chars, etc), and between these types you have functions (for instance, the function from Char to lists of Chars associating to a char c the one element list [c]).
A functor F is a rule which associates to every type T in C another type FT, and to every function f from type X to type Y a function Ff from type FX to type FY, i such a way that it behaves nicely wrt composition.
As an example, consider the list functor L. To a type T it associates the type LT = [T], the type consisting of lists of elements of type T, and to a function of types the obvious associated function between the list types.

>>7985783
From what I gather they are exactly the same, unless there is some third definition of functor which does not coincide with the one commonly used in Haskell.

To be more precise, we associate a small cartesian closed category to our functional programming language in the obvious way, and a functor in this programming language will then be an endofunctor of the associated category.

>>7985785
To clarify: the types are things like Char, Int, String. Individual strings and chars are not types.

>>7985805
>>7985785
I have a few questions will follow up after I'm free.

>>7985805
> the types are things like Char, Int, String. Individual strings and chars are not types.

Dumb question but how do Char, Int, String differ from individual strings and chars?

That doesn't make sense to me because an individual char, int, string is still a char, int, string.

>As an example, consider the list functor L. To a type T it associates the type LT = [T], the type consisting of lists of elements of type T, and to a function of types the obvious associated function between the list types.

So does that mean if I have a list L of strings: L = {"a", "b", "c"} then there is a function that maps L to LT which is a list of list of type strings: LT = {["a"], ["b"], ["c"]?

So the functor takes L to LT and applies a function to it, where L is a list of strings and LT list of list of strings?

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>>7985727

Can I use this pic as a guide on what to learn in what order if I want to more into maths?

>>7985665
Hey /sci/ I was hoping for some clarification

Im somewhat of a number cruncher in my job and some fucktard deleted my calculation of the f statistic in one of my comparison models.

Ive tried looking for a definitive answe myself, but Ive yet to track down a correct formula for replacement.

The numerator is where my issue truly lies. The denominator is simple enough; the error sum of squares in the unrestricted model divided by the sum of the number of observations minus the number of parameters.

I figure that the numerator either has to be the ESS of the restricted model divided by the number of changes to the equation or the ESS of the resricted model - the ESS of the unrestricted model divided by the # of changes.

If you could help itd be much appreciated, Ive got to file it in a week so no rush

>>7985899
>Dumb question but how do Char, Int, String differ from individual strings and chars?
Think about animals. A dog is certainly an example of an individual animal, but there is a difference between the idea of an Animal (the type) and a dog (the instance).

>So the functor takes L to LT and applies a function to it, where L is a list of strings and LT list of list of strings?
A functor is in some sense like a function itself. In this case, it is defined on types (but not on the objects itself!) and on functions between two types. Consider the example of the list functor L which takes any type T to a list [T] of that type and takes a function f: S -> T to L f: [S] -> [T] by the rule L f = map f. A functor allows you to transform types of objects into different types of objects, so you can go between say Strings and integers in a nice way.

>>7985680
>knot theory
When will this maymay die

>>7985727
I don't get this picture. Partial differential equations are somehow not "serious math?"
Sure, if you're in a course where you simply learn to plug and chuck recipes to solve boundary- and initial-value-problems of the wave equation and poisson's equation, maybe.
Same with Fourier series, they are less serious than Hilbert spaces? I think not.
And "homeomorphism" is just like a random buzzword, the way it is placed.

>>7985906

Yes. Start with one-time pad decryption.

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>>7985665

>>7985906
No. Some of the stuff at the bottom is completely made up.

>>7985920
Well I can only assume the creator of this picture has never done any functional analysis based pde / harmonic analysis classes and if you only know those concepts from your calculus lectures they might seem trivial

>>7985783
What >>7985794 said.
But it should be pointed out that e.g. in Haskell the defining algebraic rules of concepts (e.g, for functors, how Ff.Fg=F(f.g)) is not checked by the langage/compiler.
(So you can in principle go ahead and define a to be functor by some mappings while the algebraic relation might not hold and you have just random definitions.)

>>7985899
In a judgement
t:T
the left side is a term t, and the right is the type.

Trump : Human
Obama : Human
Bugs_Bunny : Animal
3 : N
7 : N
(3, 6) : N_and_another_N
(6, Obama) : N_and_Human
"hello!" : String
"mouse" : String
"k" : Char

You might have a type ListOfStrings, so that e.g.
["thread", "hello!", "mouse", "boy"] : ListOfStrings

You might have a type Human_and_Animal, so that e.g.
(Obama, Bugs_Bunny) : Human_and_Animal
(Trump, Nemo_the_Fish) : Human_and_Animal

And a functor is something that maps between types. If L is the list functor, then e.g.
L String := ListOfStrings
So that you could also write
["thread", "hello!", "mouse", "boy"] : L String

and e.g. L L L N would be the type of lists that contain lists that contains lists of numbers. E.g.
[[[3,4],[1]], [[3], [2,6] ,[3,5]]]: L L L N

And you may consider functions
N -> N
which is also a type.
E.g. the function f(n):=n+3 is of that type. I.e.
f : N -> N

And you may consider the functor that maps types to functions into a given type. e.g.
H T := T -> N

Then e.g. H N is
N -> N
and
H String is
String -> N

(That's called the contravariant Hom-functor)

But the real point of functors is that they can not just be applied to Types but also to functions


E.g. for the list functor L, and some function g : T->T between two types, you'd define a function
(H g) : H T -> H T
in such a way that it maps list_of_ts:H T to a list of things that is g applies to all entries

So for example, if we look at f(n):=n+3 again, the list functor L you'd define a function
L f
of type
L N -> L N
in such a way that e.g.
(L f)([1,7,3]) := [f(1), f(7), f(3)]

>>7985920
>>7985943
Why do you guys take it seriously, it's a troll pic you spergis

>>7985750
This shit looks like it's straight out of a random math paper generator

>>7985958
I was just commenting on it, because I constantly see /sci/ posters having exactly those opinions unironically.

>>7985958
High schoolers and college freshmen come to 4chan and can't tell what is a joke and what isn't, and then they unironically adopt stupid viewpoints.

>>7985665
How do I solve this? ⨌xyzt dxdydzdt

>>7985997
This seems to be a bit too advanced.
My guess would be to consult an expert on triple integrals and then use a quadrature for the 4th dimension.
But I'm just grasping at straws here

>>7985680
Last time I saw it the only exclusions were calc 1-3 and differential eq.

>>7985997

What you'd do is follow the steps like you would through a triple integral; while the double and triple integrals don't necessarily describe area and volume, they can be used to calculate them. I believe at four and up, hyper-volume is the correct term, in which "t" is generally time. I guess what you'd be calculating is unit of space-time.

>>7985721
Isn't triple integrals Calculus 3 lol?

please dont let this thread die i have a question that im bout to post in a sec

Let [math] \left( {M,{\mathcal{O}_M},{\mathcal{A}_M}} \right) [/math] be a smooth manifold of dimension n.

Let [math] \left( {U,h} \right) \in {\mathcal{A}_M} [/math] be a local chart on the manifold.

And let [math] {\omega _{vol}} \in {\Omega ^n}M [/math] be a volume form on the manifold.


We can define the local pullback, [math] {\left( {{h^{ - 1}}} \right)^*}{\omega _{vol}} = {g_U} \cdot {\operatorname{dx} ^1} \wedge ... \wedge {\operatorname{dx} ^n} = {g_U} \cdot vo{l_{{\mathbb{R}^n}}} [/math]

Where [math]{g_U}[/math] is some function.

Using this we can define a notion of integration on the manifold by [math] \int\limits_U {{\omega _{vol}}} = \int\limits_{h\left( U \right)} {{g_U} \cdot {{\operatorname{dx} }^1} \wedge ... \wedge {{\operatorname{dx} }^n}} \equiv \int\limits_{h\left( U \right)} {{g_U}\operatorname{d} \lambda } [/math] where [math]\lambda[/math] is the lebesgue measure on [math]{{\mathbb{R}^n}}[/math].

Now a general measure on the manifold can be defined w.r.t. the borel sigma algebra: [math] \mu :\sigma \left( {{\mathcal{O}_M}} \right) \to \bar \mathbb{R}_0^ + [/math]
such that
[math] \mu \left( U \right) \equiv \int\limits_U {{\omega _{vol}}} \equiv \int\limits_{h\left( U \right)} {{g_U}\operatorname{d} \lambda } [/math]
Now for the question:

Using this measure, how can I define a general notion of integration over the entire manifold?

its about rational algebraic expressions but i cantt seem to find a way to post it here one sec

>>7986080
that broken line of code should show:
[math] \mu :\sigma \left( {{\mathcal{O}_M}} \right) \to \bar{ \mathbb{R}}{_0^ + } [/math]

(1-(3x^2)/(1-x^2 )) ∶(x/(x-1)+1) this is as simple as i could find can anyne help me how to solve this ?

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>>7986092
this is how it looks can anyone just explain how to do this

>>7986094
2-1/x+1

>>7986133
how did you get that please tell me ?

>>7986136
Short method:

Microsoft Math, Wolfram Alpha, etc.

Slow method:

1 - Change 1-x^2 to -(x^2-1)
2 - Factorize x^2-1
3 - Realise you can simplify the fraction by multiplying both sides by (x-1)
3 - Develop
4 - Separate the numerator in two fractions and immediately reduct to equal denominator and sum them both up.
5 - Realise the fraction result has one polynomial in the denominator and other in the numerator and both have got the same degree of x so doing the polynomial long division will certainly help.
6 - Do it, phaggot
7 - Profit.

>>7986080
>>7986086
sci pls help

>>7986077
You realize threads on sci are on 30 hours after the last post

>>7986080
there are two ways. don't integrate over a whole chart, integrate over polytopes that are subsets of the charts and then partition the manifold into "disjoint" polytopes (their boundary can overlap) and sum it. or you take the whole atlas, integrate every chart and then you average it using partition of unity. the first method is more transparent, but it relies on the fact that you can triangulate every manifold. the second method is easy to define and works like a charm in proofs but is almost impossible to use for actual computation (partition of unity...)

>>7985915
Thanks this clears things up!

>>7985946
Great explanation, written to be very thorough and quite accessible. I understood everything you were saying up to:

>And you may consider the functor that maps >types to functions into a given type. e.g.
>H T := T -> N

and below.

I do not understand the H T: T->N notation. It looks like T->N is mapping a type to a function whose domain is the Integers. I don't know what HT means though, but according to your earlier definition HT is the term and T->N is the the type, but again HT itself-- dunno what that represents.

Again, don't understand HN.

Lost me on " contravariant Hom-functor"

Maybe once I understand those things I'll understand the rest of what you wrote.

>>7986426
>>I do not understand the H T: T->N notation. It looks like T->N is mapping a type to a function whose domain is the Integers. I don't know what HT means though, but according to your earlier definition HT is the term and T->N is the the type, but again HT itself-- dunno what that represents.

Given any two types S and T, there is a Haskell type S -> T of functions from S to T. Pick N to be absolutely any type you like (I'll use Int here for clarity, but think about the more general N after it clicks). Here H is our functor, and it takes a type T to the type T -> Int. Here H T is simply applying the functor H to some type T -- we are defining that H will take any type T to the type T -> Int. Now that you know how it is defined on types, think about how to define it on functions; this is an extremely common functor.

>Lost me on " contravariant Hom-functor"
It's an important thing to read about if you want to give the basics of category theory a serious look at some point.

>>7986453
Okay this makes more sense now.


You wrote:

"And you may consider the functor that maps types to functions into a given type. e.g.
H T := T -> N

Then e.g. H N is
N -> N
and
H String is
String -> N

(That's called the contravariant Hom-functor)"

What is the contravariant Hom-functor here? Applying a functor H to strings that maps type strings to type N? Is it simply applying a functor to a mapping that maps from one type to another?

A contravariant Hom functor is just any functor like the H defined above. It's called this because in category theory, we use the notation Hom(T,N) to mean the type T -> N. Contravariant Hom functors are nice because given a function f: S -> T, we get another function H f: (T -> N) -> (S -> N). This is saying that we can not only transform types, but we can take any function from T -> N to an associated function S -> N.

>>7986426
I've written
>H T := T -> N
and you seemingly read it as
>H T: T->N

T is any type, N is the type of integers, T->N is the type of functions from T to N, and H is a functor that e.g. maps the type T to H T, which in the example is T->N. Note that if T is a type and N is a type, then T->N as a whole is also a type. The type of functions from T to N.

>>7986540
H is the functor. It maps types (e.g. T) to types (here H T) and functions between types (e.g. f:U->V) to functions between types that are the image of H (here (H f): (H U-> H V), i.e. where H f is the functions from H U to H V)


Btw. that post between the two posts I quoted, that was another anon.

>>7986589
Two minor, almost autistic, comments

>This is saying that we can not only transform types, but we can take any function from T -> N to an associated function S -> N.
The transformation of types comes already with the functor, but the association you mention is one associated with a term you pass.

>It's called this because in category theory, we use the notation Hom(T,N) to mean the type T -> N.
The endofunctor mapping T to T->S is rather the internal hom function hom(-,S) than Hom(-,S), if by the latter we understand the set valued functor (the historically earlier notion.)
The fact that in Set we can consider it an internal hom functor Hom(S,T)=T^S is nice but not true for all categories C (not all have exponential objects like that, while the hom-functor Hom will still exists, just not map back to C)

>>7986426
Read Maclane you troglodyte

I'm a freshman pleb still in Cal II. Just wanted to say that reading this board inspires me. Hope you all have a good day/evening/etc.

So I am confused about thr dimension of manifolds
I mean I have something like (everything in IR ofc):
M={x^2+xy-y-z=2x^2+3xy-2y-3z=0}
A 1-dimensional Manifold (!)
And
N={x^2+y^2-z=0} a 2-dimensional (!)
Why couldnt M be a 1-dimensional Manifold
I just dont see where these 2 sets differ in a certian way

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>tfw I got an A on my master thesis

>>7987451
Since i've never taken any classes on differential geometry or manifolds, take my answer with a grain of salt.
The difference is the degrees of freedom.
For M you have two equations for 3 variables, for N you only have one.
Example: N describes the exterior of a cone which is a 2d manifold.
If you add the condition z=1, you get a circle (1d)

>>7987475
Oh, that is reasonable with the definition I got
Thanks very much!

>>7987451
Generally, adding in a constraint like that will always lower the dimension by 1 in math. Think about linear algebra.

>>7987486
Just to add to that. Don't take this heuristic of m equations and n variables = n-m dimensional manifold too serious.
It's something that works in linear algebra when you have a system of equations without redundancy, but once you go nonlinear you can't really make such broad statements anymore

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Any recommended introductory texts on SPDEs?

I studied SDEs and PDEs and now I want to go deeper.

>>7987552
Beautifup thing is that by differentiating, the 'heuristic' just becomes a theorem.

>>7987451
The dimension of the manifold is determined by dimension of the euclidean space the charts map too.

i.e.
If you have a manifold [math]M[/math] with an atlas [math]{\mathcal{A}_M} = {\left\{ {\left( {{U_\alpha },{h_\alpha }} \right)} \right\}_{\alpha \in I}}[/math] where [math] {h_\alpha }:{U_\alpha } \to {\mathbb{R}^n} [/math] , then M is n-dimensional.

>>7985665
>No Calculus I-III, Differential Equations or elementary Linear Algebra questions allowed.

Excellent because I have a Pre-College Algebra question:

What does 1*2+3-4(5+6)^7 =

How can I figure this out when it's got so many different types of math in it?

>>7988082
what is a dimension? what is a manifold?

>>7990024
SADMEP

subtract then add then divide then multiply then exponentiate then print the final results

1*2+3-4(5+6)^7
=1*2+(-1)(5+6)^7
=1*1*11^7
no division
=1*11^7
=1*11^7
=11*11*11*11*11*11*11
=121*11*11*11*11*11
=1331*11*11*11*11
=14641*11*11*11
=161051*11*11
=1771561*11
=19847171

e.z.p.z

>>7987582
Honestly man, if you want to study SPDEs just read papers. I highly recommend some of Davar Khoshnevisan's papers. They are easily accessible and well written.

Read Martin Hairer's books on rough path theory as well and if you're feeling ballsy read his landmark paper on regularity structures. That is currently beyond me.

SPDEs are an active research topic.

>>7987464
Do you get a grade on a thesis? Wtf? I just went straight into PhD and they just approve your thesis or reject it.

>>7990057
lol nice try, but I know it is PEMDAS nothing is is more important than parentheses!

>>7990057
Serious question:
If we changed the order of operations, how would that affect high-level math? Could it make entire theorems no longer work, or could proofs and formulas just be rewritten?

>>7990137
They would be rewritten in an awful longwinded way.

>>7990137
Wouldn't change anything serious.

Also O of O isn't arbitrary. Operations are ordered to represent complexity.

>>7990137
Are you retarded? You would just have to use a lot of parenthesis. The order of operations are just syntax.

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>>7985750
Obviously, have you even read IUTchII? Think before you post and embarrass yourself with such obvious questions. Fuck.

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Alright you fucks. I have a physical model [math]\mathcal{L}[/math] on a manifold with boundary [math]M[/math] and a symmetry [math]G[/math] of [math]\mathcal{L}[/math]. I know the generators of the Lie algebra [math]\mathfrak{g}[/math] of [math]G[/math] and that the manifold [math]M[/math] requires three patches [math]U, U_\pm[/math]. I have written down the Chern-Simons term on the interior and the boundary of the model
[eqn]
\int_{\partial U}\operatorname{tr}\left(AdA+\frac{2}{3}A^3\right) + \sum_{\sigma = \pm}\int_{\partial U_{\sigma}}\operatorname{tr}\sigma\left(A_\sigma dA_\sigma +\frac{2}{3}A_\sigma^3\right),
[/eqn]
the goal is to use this to characterize the fractional quantum Hall effect. However, under gauge transformations the Chern-Simons terms give
[eqn]
\int_{\partial U}\operatorname{tr}\left(g^{-1}dg\right)^3 + \sum_{\sigma = \pm}\int_{\partial U_\sigma}\operatorname{tr}\left(g_\sigma^{-1}dg_\sigma\right)^3
[/eqn]
where [math]g, g_\pm[/math] are the coordinate transition maps on the three patches. Now [math]M[/math] is 3-dimensional so these gauge transformations are characterized by [math]\pi_3(G)[/math], which I'm inclined to believe is [math]\mathbb{Z}[/math], which doesn't contain any fractions.
How do I get fractional topological quantum numbers from Chern-Simon terms?

>>7985727
I thought OTP decryption was unbreakable by definition because no information could be extrapolated about the plain text from the cipher?

see
>>7992777
I'm trying to see if I can make a non-stepped physics simulation algorithm but I'm wondering what it is with maths that means you can't calculate 3-body problems.

>>7985665
After 4 years, maybe more if you are a grad students, of math study are any of you able to solve IMO and Putnam questions?

Is an IMO high schooner as good or better than a math major? a phD?

Is 1.111111111111111111111111.... = 10 in base 2?

>>7993419
yup, it's equivalent to the sum over 1/2^n which coverges to 2

>>7993579
How out of touch with people are you to think that is considered a clear explanation?
While the answer makes sense to me, answering a simple question with advanced math is silly because the answer will be more confusing than the question.
Just say that since 1 is the highest digit in binary, and 9 is the highest digit in decimal, asking if 1.111111111111 = 10 in binary is like asking if 9.9999999999 = 10 in decimal.

>>7993288
From what I heard IMO are good at solving problems not typically seen in undergrad math, ones that require clever insight. Eastern European schools seem to teach their students towards this kind of thinking while Western counterparts have a more traditional method of teaching math. From what I've heard/seen having a BS in math will not necessarily translate to being able to solve IMO problems

>>7993275
the three body problem can only be solved analytically with additional conditions.
It's not really special in that way - Most differential equations cannot be solved analytically and we can only make statements about existence, uniqueness, asymptotics or linearisations of the problem.
The rest has to be done numerically unfortunately

>>7993587
You are in an advanced math thread, my friend.
Since the OP excludes people without calc knowledge, it's perfectly acceptable to give an explanation with a geometric series.
If you want to continue baiting, just remove yourself from this thread and open up another 0.99.. =/= 1 thread

>>7993591
I gave an intutive explanation without using any advanced math, which is even better than using a geometric series.
Also, I didn't imply 0.99...=/=1. I was saying that it's like asking if 9.9999999999 = 10 in that once you realise it's the same question, the answer is obviously yes.

>>7993595
>geometric series
>advanced math
Your explanation is pretty shit btw, since you didn't even manage to answer the question.

>>7993138
What do you mean by L? Is it a function from your manifold to R? What do you mean by a symmetry G? Is it a group with an action on {L = 0} or something? Is it a Lie group, and is the action required to be smooth? Etc.

Can anyone recommend a reference for learning about curvilinear coordinate systems? The professor is an asshole and can't teach.

In grade school arithmetic to multiple two numbers x and y together it takes 2*n^2 operations to do the multiplication and carry overs. But I saw it takes another 2n^2 operations to do the additions from those results. How is that so?

>>7985727
>>7985906
That image is a meme and very poorly indicative of the different fields of math.

Why isn't there a quaternionic analysis like with the complex numbers and the reals

>>7993196
>no information could be extrapolated

As far as we know.

>>7993607
it's a Lagrangian function (the integrand) and is invariant if its argument are changed using the G-group action.

>>7993138
not that you'll get an answer here, but how are the fractional quantum hall numbers defined?

To help us help you, write down the full computation in case you ones doesn't need several patches.

>>7993686
It's a normed division algebra, I'm sure you can do a lot of things,
but C is already algebraically closed and Taylor series of non-commuting objects is a little obscure. Sure you have it in the theory of operator algebras on Hilbert spaces, though. The quaternions are akin to su(2) anyway.

>>7993686
There is. It's just not popular.

In the general definition of the Néron-Tate height [math] \hat{h}_L(P) [/math] (https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Tate_height) of a point [math] P [/math] on an abelian variety [math]A [/math] we need some symmetric invertible sheaf [math]L[/math] on [math]A[/math].

I've learned the specific case from Silverman's book where [math]A/K[/math] is an elliptic curve defined over a field [math]K[/math] and [math]\hat{h}_f(P)[/math] was defined from some even function [math]f∈K(A)[/math]. How do I reduce from the invertible sheaf point of view to this specific case? I'm having trouble seeing this sheaf.

Is there a simple argument as to why [math]x^2(x-1)^2 + y^2(y-1)^2[/math] is irreducible in [math]\mathbb R[x,y][/math]? I have a very long winded proof that I don't like.

>>7985919
its literarly just one grad student who studies it and reposts it every now and again on /sci/

>>7993710
They aren't defined, they're observed. However it's conjectured that the quantum Hall effect (both integer and fractional versions) is a topological phenomenon, since integer quantum Hall effect can be fully explained by magnetic fields threading the sample, which creates a holonomy group isomorphic to [math]\mathbb{Z}[/math] on the principal bundle [math]P = P(M,G)[/math]

>>7994256
If there's only a working notion of those numbers, then I'm afraid noone who doesn't actually already know how to do it can help you

>>7994217
what's a sketch of your proof?

>>7994326
It can't have a linear factor as an element of [math](\mathbb R[x])[y][/math] because then there would be more than 4 real roots. So we assume a factorization of the form [math]\displaystyle \big(y^2 + g_1(x)y + g_0(x)\big)\big(y^2+h_1(x)y+h_0(x)\big)[/math] and notice that there are only 4 possibilities for what [math]g_0[/math] and [/math]h_0[/math] can be. Then you make some arguments based on what the constant terms of each polynomial would need to be and show it can't work out this way.

>>7994350

Assume this factors as g(x,y)h(x,y). As you say, no linear factors. Without loss of generality let the coefficient of x^2 in g(x,y) be 1. Then h(x,y) has the same coefficient 1 in front of x^2. Let the coefficient of y^2 in g(x,y) be c. Then the coefficient of y^2 in h(x,y) is 1/c. But the coefficient of x^2y^2 in g(x,y)h(x,y) is c+1/c which is supposed to be zero.

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If this ssurface were to modified such that it becomes compact, would it be possible to use Van Kampen's theorem to find it's fundamental group.

>>7985680
>If you haven't learned what I learned two semesters ago you are nothing

>>7985727
>irrational pattern functions
gets me every time

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dont know if you guys can handle this, it's pretty tough

>>7994968
You can figure it out by just looking at it bro.

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>>7985665
>Differential Equations
This too hard for you op?

>>7995125
the OP clearly means no calculus based ODEs. analysis based is fine.

for example your idiot plug and chug problem is not acceptable

>>7995125
What's with that shit formatting

>>7995143
>for example your idiot plug and chug problem is not acceptable
That problem is from a final in a grad level ODEs course.

>>7995228
There are no 'grad level ODE' courses unless you go to a community college or something. That's like saying 'grad level calculus'. You should be doing PDEs by this point at any respectable institution.

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How do ordinals & infinite sets relate to well-ordering principle when you deal with infinite sets such as the reals? In other words what is the connection between omega 1 and well ordering?

What confuses me is the reals have no least element so is set that contains the ordinal omega 1 well ordered?

This confuses me

When does it make sense to talk about well-ordering when dealing with infinite sets?

>>7985665
>Diff Eq and Linear Algebra
>Advanced

>>7995417
>There are no 'grad level ODE' courses
nonsense

>>7995228
That is most definitely not grad level

>>7985727
>implying von neumann couldnt solve any of those problems deemed beyond human ability if given enough time.

>>7995420
if you accept choice (which you should) you can always well-order a set

so you can give a well-order to R as a set. it's just very different than the usual order on R and isn't compatible with its operations. so it's not really R anymore

>>7995417
>>7995448
I'm embarrassed to say this, but it looks a little like my first year grad engineering math course.

I don't know how planners get away with calling a class advanced, when they rehash your intro to the subject you took 2 years before with a couple extra algorithms.
Oh so you added a diagonalization method to my lin alg class....why wouldn't I have understood matrix multiplication last time?

>>7995228
You literally have to plug and chug. This is a shit program, not even math

>you need to argue your answer

fucking KEK. is this a gender studies grad course on ODEs or something?

>>7995465
>engineering
>math
there's your problem. if you want math, take the math version.

>>7995462
I'm confused by you saying you can well-order R, how?

So, how does the well-ordering principle apply to transfinite numbers past alpeh null?

does it make sense to say the ordinals 0, 1, 2, ... omega, omega+1,... is well ordered even though it's an uncountably infinite set?

sorry for the dumb questions trying to wrap my head around it

>>7995486
a total ordered set is called well-ordered if every subset has a minimum element

so yeah, that set is well ordered, infinity really doesn't matter. if your order is properly arranged like with ordinals, it has the well order property

>>7995473
The math version doesn't count to my degree...
Yes I went to school to get qualified to get employed.
If I wanted to learn something I'd read a book.

>>7995493
>I don't know how [they] can get away with calling a class advanced and [making it shit]
>If I wanted to learn something I'd read a book.

They get away with it because engineers, like you, are dumb, lazy faggots who don't care about learning. Why the fuck are you complaining or trying to talk about math if all you care about is going to do a repetitive job and getting some money?

>>7995491
ok, so the set with the 0, 1, 2,..., omega, omega+1 is well-ordered because it has a minmum element

the reals is well ordered because every subset has a least element

ok if what i said above is true then that helps a lot!

if i said anything false please correct me... believe you answered my question with the subset part

>>7985967
Good. That's the only way we can discriminate between the wheat and the chaff.

>>7995503
every subset of R doesn't have a least element. take for example the set

[math] S := \{ {x \in \mathbb{R} / x*x > 2 } \} [/math]


R with its usual order is NOT well ordered. if you want to well order R as a set, you need to give it a new order, one that looks like the one for the ordinal w_1.

it could be very weird, something like
pi < e < 1 < 9 < 4 < ... < ... < -1/2

>>7995503
This is totally wrong, as is what the other guy said to you. The reals under the usual order are definitely not well ordered. What's the minimum element of [math](, \infty) [/math]? The well ordering theorem says you can find some order under which R is well ordered, but good luck figuring out what it is.

>>7995502
So I can afford books on math and not live with my parents like you
:^)

>>7995524
Ok, I see what you mean. So if you want to well-order R with the w1 example then you'll have to stick some real number as the "first" ordinal followed by your successors... how the fuck does that even work? how do you choose the "first" real for your first ordinal?

Initially I thought the 0,1,2...,w1,... set wasn't well-ordered with the reals, but now i see how it is if you choose some "first" real to be your first ordinal... just confused what the fuck that looks like

>>7995529
so if you rearrange R, you can make it well ordered.... and hence the 0,1,2,...,,w1,...,e0 is well ordered? - that is mostly what i am wondering about, if that list is well-ordered even though its uncountably infinite

>>7995537
well it looks just like w1.
only the elements have other "names"
but any well ordering is isomorphic (as in, the same save for names) to an ordinal. so it should come as no surprise

if you give R an order like that it's no long compatible with its operations. for example you no longer have
a < b => a+c < b+c
etc etc. might as well be any other set, it stops being R at that point

>>7995503
Here is an exercise:
A set R totally ordered by a binary relation [math] \preceq [/math] is not well ordered if and only if
there is descending sequence [math] a_0\succ a_1\succ a_2\succ\dots[/math]
So the reals with the usual order are not well ordered because you can find a descending sequence -1,-2,-3,etc. The nonnegative reals (positive reals together with the 0) has a minimum (0) but is not well ordered either, because you can find a descending sequence
[math] 1>\frac{1}{2}>\frac{1}{3}>\dots[/math]
If you try to find a descending sequence in [math] 0,1,\dots,\omega,\omega+1[/math] you'll hopefully become convinced it is impossible.

>>7995555
Check em!

Okay, this makes more sense.

Last question for now...

How is 1,2,3,...,w,w+1 well ordered if there are more than alpeh0 ordinals there?

>>7995562
your question doesn't make sense senpai
>how is 1 a number if there's an infinite amount of numbers?

>>7995562
I don't understand the question.
I mean that's similar to saying how can 3 and 7 be natural numbers when there are a whole lot more natural numbers.

>>7995565
1 here is the first ordinal in my infinite list...

I thought you said the ordinals 1,...,w1 was considered well-ordered?

>>7995568
i mean is the ordinal numbers from 1 to w+1 and beyond well-ordered?

>>7995569
it's well ordered, every subset has a smallest element

what does it having more than aleph0 ordinals have to do with anything?

>>7995577
Well the set of ordinals I'm talking about is larger than aleph0, which is uncountably infinite. im getting confused because you said every subset of that ordinal list has a smallest element, but uncountably infinite sets (such as reals) don't have a smallest element? im just in a confused fuck state

>>7995583
>uncountably infinite sets don't have a smallest element

that's where you're wrong. just because R with its usual order doesn't have a smallest element doesn't mean no infinite set can have one. the ordinals are well-ordered by construction, which means they do have this property. you have arbitrarily (well, kinda) large sets with smallest elements

>>7995530
>not being a trust fund baby

>>7995585
ok, it makes sense countably infinite sets have a least element, but what uncountably infinite sets have a least element?

it's starting to make a bit more sense that ordinals are well-ordered by construction. that means we have to construct uncountably infinite sets to obey this property of having a least element?

thanks for taking the time to unconfuse me

>>7995576
Two things:
Ordinals usually start at 0.
You may have heard that the set of all sets doesn't exists (Russell paradox). The set of all ordinals doesn't exists either for similar reasons.
But if you take an ordinal [math] \alpha [/math]
the set [math] 0,1,\dots,\beta,\dots[/math] of ordinals [math] \beta<\alpha [/math] is well ordered (by the way this set is identified with [math] \alpha [/math] in the Von Neumann (the usual) definition of ordinals, so you can make the ordinals by a recursive process starting with
0=empty set, [math] 1=\{0\}, 2=\{0,1\},3=\{0,1,2\}, etc, \omega=\{0,1,\dots\}, \omega+1=\{0,1,\dots\}\cup\{\omega\}, etc,etc,etc [/math]

>>7995592
it turns out all well-orderings are essentially the same. that is, if you have two well-ordered sets (they fulfill this "all subsets have minimum element" property) of the same size, then they're the same order with the names changed.

so, since we have an ordinal for each "size", all well-orderings are essentially just ordinals

>>7995593
you can't really recurse that long though, can you? it gets tricky. you can get past finite ordinals with the axiom of infinity and past the next few ones with the power set axiom, but what after that?

it's very fuzzy for me. do you know anything about big ordinals?

>>7995604
Yes you can, hence the name "transfinite recursion". There are loads you can say about big ordinals, sort of like you can say stuff about unimaginably huge finite numbers for example https://en.wikipedia.org/wiki/Ackermann_function.
Any way, the recursive "definition" of ordinals is circular (you need ordinals to do recursion) but it gives a strong intuitive picture to guide you on your reading of the full rigorous details in any set theory book.

>>7995595
So you can have uncountably infinite ordinals and it'll still be well-ordered? e.g.,

0, 1, 2, . . . , ω, ω + 1, ω + 2, . . . , ω + ω =: ω2, ω2 + 1, . . . , ω3, . . . , ω2
, . . . .

because every nonempty subset has a smallest element

eg: O, {1} , {1, 2} , {1, 2, . . . , n} , {1, 2, . . . } , {1, 2, . . . , ω} , {1, 2, . . . , ω, ω + 1} .

My question is when you have an uncountably infinite ordinal list like this what do the uncountably infinite sets those ordinals represent look like? it confuses me how an uncountably infinite set can be constructed to have a least element. this is my confusion

>>7995612
I don't know what transfinite recursion is, though I know what transfinite induction is. I have the intuitive knowledge: I took my set theory class long ago. I'm actually a math junior. But I don't know anything about classes and categories, the furthest you can go with just the barebones set theory is using infinity + power set to build up to a point.

Where can I read about this?

>>7995624
Transfinite recursion is just definition by transfinite induction. To refresh knowledge of set theory the first chapter of Kunen's "set theory" is pretty good.

>>7995633
you can't define something with transfinite induction, at least not within set theory. you need a set first.

unless you're working from a bigger class and then dropping down?

Can someone address >>7995614

I still don't understand how something uncountably infinite is well-ordered.

>>7995635
By "definition by transfinite induction" I meant sort of like the process you use to define stuff using induction in the natural numbers, for example [math] (n+1)!=n!\cdot n [/math]. There is a technical snag in that recursing over the ordinals may give you a "proper class" instead of a set, but for these details there is no substitute to just reading a book (like Kunen's)

>>7995614
>>7995638
I don't know what you're saying here bro. The sets are just what they are. What they "look" like is sketchy at best, maybe a dotted like that kind of gets closer and closer and then jumps to a bigger infinity every few inches? who knows

>>7995645
ok then whatever the sets looks like doesn't matter, but when you say every subset has a least element do you mean every ordinal has a least ordinal? which is one reason ordinals are well-ordered no matter how "uncountably" infinite they may be?

>>7995644
you can't recurse from a set and build a class. you need to have been working from the class since the beginning

>>7995650
a least ordinal?
a set of ordinals can be something like {2, 3, w} which has a smallest element 2.

any such set of ordinals, as weird as it can be, has a smallest element. that's all

>>7995654
ok cool.

well correct me if i'm wrong but what i got out of all of this is all ordinals (finite, countably infinite, uncountably infinite) are well-ordered.

>>7995651
There is a rigorous definition for "proper class", you can start from a set and get a class because the ordinals are a proper class themselves. Go read the book, man.

>>7995655
yes

>>7995663
thanks bro. thanks for taking the time to explain this and un-confuse me.

>>7985727
Is it even mathematically possible to decrypt a one pad cypher?

>>7995675
long as key isn't known or reused then no.

What the fuck is the motivation between Solobev spaces?

>>7995685
I don't know about motivation but you at least need them to prove some theorems like Hodge decomposition.

>>7995685
having weak derivatives & solving PDE

>>7995420
the reals are not a set, but a space and size are not well defined beforehand

>>7995685
We have a neat theory of self-adjoint operators. So it makes sense to look for spaces where things like the Laplacian become self-adjoint.

Hello Anons, could anyone explain why would a=b(x+yi) equivalent to a=bx, where a,b,x,y are all integers, but x+yi is from Z[i]? I found this while looking at the proof that b/a in Z <=> b/a in Z[i]

>>7995768
> space
You mean field?

The idea of defining a regular function on a variety as being locally a quotient of polynomials has never really sat well with me. I think part of the issue is that I'm not certain why we want rational functions rather than just polynomials. Can anyone shed some light on how they think about this?

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I feel really stupid for asking these. I have the following formula for trace

[math] \text{Tr}_K^E(\alpha)=\tfrac{n}{t} \sum_{i=1}^t \phi_i(\alpha) [\math]

where [math] \phi_1,...,\phi_t [\math] are the distinct K-homomorphisms from E into the closure of K.

I feel like both of these should immediately follow from t<n or t=n respectively. The only thing I got from Google/stackexchange was a quick proof using Dedekind's lemma, but we never covered that in our class. Can some kind anon at least provide me with a hint?

>>7997302
Because if a,b,x,y are integers, y has to be 0 for that equality to work.

a, b, and c are the sides of a triangle, while A, B, and C are the angles opposite each side.

Why does (a - b)/(a + b) = [tan 1/2(A-B)] / [tan 1/2(A+B)?]

>>7997340
[eqn] \text{Tr}_K^E(\alpha)=\tfrac{n}{t} \sum_{i=1}^t \phi_i(\alpha) [/eqn]

>>7997347
that's how I tried to do the step, but my colleagues say it's not true, so I am not sure. Do you know if this fit's correctly in that proof?

>>7997384
If [math]y[/math] is nonzero (and [math]a[/math] too, I guess) then [math]a[/math] isn't even a real number, let alone an integer.

>>7997340
For the first problem use the formula for [math] F/L/K [/math] extensions to reduce to the purely inseparable case
[math] L=K(\alpha) [/math], with [math] \alpha [/math] a p root of an element in K.

For the second problem the only way I can see how to prove it is by remembering that the bilinear form [math] \text{Tr}_K(\alpha\beta) [/math] is nondegenerate, (so the problem becomes trivial). To see that it is nondegenerate get a primitive element (by the primitive element theorem), a, take the matrix of the bilinear form in the basis the powers of a; and use the formula you mention to express the matrix as a the product of a Vandermode matrix with its transpose.

>>7997389
a+bi=c(x+yi) is equivalent to a=cx and b=cy can I use something similar to derive that?

>>7997406
a,b,c,x,y all in Z, b shouldn't be 0 I think

>>7997320
I think of functions in general as rational functions (or things that can locally be written as rational functions) and of regular functions as functions that do not have poles inside the open set I am considering.
In some cases it might be polynomials (for example on the affine line), but on smaller domains (for example the affine line minus the origin), this allows for rational functions as long as they don't have poles: for example, on A^1\{0}, you can have any rational function whose denominator is a power of x.

>>7997422
This is nice. Is there a way of continuing this line of thinking over a non-algebraically closed field? For instance 1/(x^2 + 1) has no "poles" in R, but I'd want the global regular functions to just be R[x]. My guess would be that leaving the realm of varieties forces you to give up these nice characterizations.

What is shit like the Mean Value Theorem and Newton's Method actually used for?

>>7997608
>Mean Value Theorem
Proving Things
>Newton's Method
approximations

how do I solve lim 2^(1-x) as x->1, I know the solution is 1.

>>7998604
also the same when x -> infinity

>>7998604
>>7998605
oh solved it I think, i need help with [math] \sum\limits_{n=1}^N n^{-k} [/math] k < 1

>>7985759
A clear wildburger influence

>>7998604
a^0 =1 for all a dude, how is that even a question

>>7997319
>>You mean field?
rather a space as a topos or a locale, not a topological space.
I found this post which try to characterize R.
http://math.stackexchange.com/questions/839848/category-theoretic-description-of-the-real-numbers

IT is odd that R has no universal property categorically, beyond the stuff about R is the terminal archimedean field


constructively-predicatively, R is not really a set, but sheaf topos, or rather a locale whose generalized elements are indeed real numbers.
https://ncatlab.org/nlab/show/real+numbers+object
here is how a theory of *one* real number is expressed
https://books.google.fr/books?id=LkDUKMv3yp0C&pg=PA367&lpg=PA367&dq=archimedean+constructive+mathematics&source=bl&ots=at40ELOwgl&sig=SdDpMvxOX2VHtaCJSBvG5qR4zA0&hl=en&sa=X&redir_esc=y#v=onepage&q=archimedean%20constructive%20mathematics&f=false

>>7998623
k>1 my mistake

>>7998659
You also need to at least mention that [math]2^{-x}[/math] is continuous at 1.

>>7998802
Or that a limit of an empty diagram is a final object.

>>7985665
Because DEs aren't important anyway :^,)

>>7998807
What does it have to do with final?

>>7998807
why does category theory even use the term limit for category theoretical limits
it doesn't even make any sense calling them limits

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>>8000047
sure they do, they are the closest/fullest object together with arrows over the diagram.

E.g if you take, in Set, the trivial diagram with two dots X and Y, mapped to A=F(X):={1,2,3} and B=F(Y):={2,3,4} then e.g.
with N={2,3} and the obvious injections

i_{NA}: N -> A
i_{NB}: N -> B

then the limit

L = A x B = {(1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}
with left and right projection
p1: L->A
p2: L->B

is the fullest object hovering over A and B:
E.g. in the above case, i_{NL} defined by 2 to (2,2) and 3 to (3,3) makes
i_{NA} factor as p1\circ i_{NL}
and
i_{NB} factor as p2\circ i_{NL}

L is the fullest, because can at most need |A|·|B|=9 elements to inject any sets mapping to A and B into L.

It's the limit in the search for a single object naturally mapping into the diagram image (A and B here), that's as near to them as possible.

Here is an example for a much more intricate limit, and you can see how it's the end of a process of doing something
http://math.stackexchange.com/questions/38517/in-relatively-simple-words-what-is-an-inverse-limit/38522#38522

>>7997349
This follows from re-writing the expression on the right in terms of sines and cosines and using sum and half-angle formulas to first show that the question reduces to showing that

[math] \frac{a-b}{a+b} = \frac{\sin(A) - \sin(B)}{\sin(A) + \sin(B)} [/math].

Now multiply out both sides, cancel out terms, and it reduces to showing that

[math] 2 a \sin(B) - 2b \sin (A) = 0 [/math].

But by the law of sines, [math] a \sin (B) = b \sin (A) [/math], so this equality is true: it is [math] 0 = 0 [/math].

Working backwards along all of this from the equality [math] 0 = 0 [/math], we derive your stated formula.

I'm currently working through Nakahara's book and I've hit a snag.

Consider the real line bundle [math] L [/math] on [math] S^1 [/math] [math] \left ( L,S^1,\pi,\mathbb{R}, G \right ) [/math]

[math] L [/math] is either the cylinder bundle or the Möbius bundle.

Consider the Whitney sum [math] L \oplus L [/math]

If [math] L [/math] is the cylinder bundle, ie [math] G [/math] is the trivial group. It is straight forward from the triviality of the cylinder bundle, that the Whitney sum is a trivial bundle as well.

However in exercise 9.1 in Nakahara's book, it asks to show that the Whitney sums of both of these line bundles are trivial.

The Möbius bundle is clearly nontrivial, and when I went to establish the transition functions on [math] L \oplus L [/math] I found (which seemed fairly intuitive to me) that the sum was also nontrivial. Was my calculation incorrect or have I misinterpreted the question?

The original question was:

"Let L be the real line bundle over S^1 (i.e. L is either the cylinder S^1 × R or
the M¨obius strip). Show that the Whitney sum L ⊕ L is a trivial bundle. Sketch
L ⊕ L to confirm the result.
"

>>8000768
ask stackexchange lad the highschoolers here wont know it

>>8000801
There was at least one guy posting earlier in the thread that knows enough about fibre bundles to answer my question.
see
>>7993138
and
>>7994256

>>8000768
I have no intuition for what to make with the groups and I've never done such an exercise.
Can you write down your construction?

In the end, is what you try to find a section that never flips sign?

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>>8000768
When you Whitney sum two Moebius strips together you get a fibre with two twists.

>>8000834
I wonder what I'm doing wrong when trying to calculate the sum explicitly then.
The transition between elements of my open cover wind up being the identity and -identity just like in the standard mobius bundle.

Im using

[eqn] T_{ij}(\theta): \binom{x}{y} \mapsto \binom{t_{ij}^L (\theta) x}{t_{ij}^L (\theta) y} [/eqn]

where am I fucking up?

>>8000856
If I understand your notation, what happens when [math]T_{ij}[/math] is the identity and when [math]T_{ij}[/math] is minus the identity? You should get a cylinder and an "upside down" cylinder.

>>8000877
Nakahara writes,
"If all the transition functions can be taken to be identity
maps, the fibre bundle is called a trivial bundle"
This was the method I was aiming to use to prove the triviality of the sum. The fact that one of the transition functions is not the identity means that the bundle is not trivial right?
This is the center of my confusion

>>8000887
Thinking about it again this doesnt seem to always hold.
In the case of a principle G bundle, the structure group is G and the transition maps are dictated by the structure group which in most case wont be identities.
In this case P(G,M) is trivial if P admits a global section. This is what I need to test for the whitney sum then?
I would just need to see if the associated principle bundle has a global section.

in general when you want to prove f(n) is big-oh of g(n) what is the approach to take to find the n0 and C?

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how would you calculate the surface of your erect penis?

>>8000856
I don't know what the hell you're talking about, but

ABC
D E
CBA
+
CBA
D E
ABC
=
ABC
D E
ABC (Cylinder)

ABC
D E
ABC
+
ABC
D E
ABC
=
ABC
D E
ABC (Cylinder)

>>7985906
Not a bad idea. You should know everything before serious math.

But I don't think you can get into serious math by picking stuff out of a list of 100 topics. For instance, you're gonna want to start with some algebra before anything else. And before algebra you need to be familiar with proofs and proof strategies.

But now that I think about it, algebra opens a lot of doors. So learn some modern (abstract) algebra, then start picking stuff from the 'serious' section that interests you.

>>8000923
I forgot about this book, lel.

Anyway, a trivial fibre bundle is a just cartesian product, that's the whole point in fact.

for the mobius strip, you have
https://en.wikipedia.org/wiki/Fiber_bundle#Nontrivial_bundles

take the mobius strip+ projection on the circle is non trivial fibred bundle, the fibre is some real interval whose structure group has two elements: the identity and the reflexion.

=>your math is right it seems

check the errata for this book...

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Guys I'm confused, why are half the posts in this thread the same as in this?
http://zeekyforums.com/thread/7985665/science/advanced-mathematics-question-answer-thread.html

>>8002173
That's a website that "archives" 4chan content without the pictures and presents it as a fake forum with a ton of ads for $$$

>>8002878
The fuck? People actually post on it too, it's not all just from sci

>>8002878
how do you earn money ? who wants to pu adds in this site?

irreducible polynomial is the same as prime polynomial, right?

>>8004607
If you mean polynomial ring over a field and don't count the 0 polynomial, then yes. This is true in any UFD, in fact.

>>8004621
Thats what I needed to be sure of, thank you!

>>7985665
What do when Lax-Milgram fails? Anything else allowing me to guarantee unique solutions for boundary problems?

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Pic is was a problem on my last HW assignment. The problem itself is fairly straightforward if you know your linear algebra, however we went over a geometric result of it in class the next day that I don't fully understand.

[math] {\operatorname{Alt} ^2}{\mathbb{R}^4} \cong {\mathbb{R}^6} [/math]. So therefore if you look at a kind of plane with [math] \operatorname{Alt} _ - ^2{\mathbb{R}^4} [/math] as the "x-axis" and [math]\operatorname{Alt} _ + ^2{\mathbb{R}^4}[/math] as the "y-axis", you can see that the set [math] C = \left\{ {\omega |\omega \wedge \omega = 0} \right\} [/math] forms a 5 dimensional cone. This is because using the decomposition [math] \omega = {\omega _ + } + {\omega _ - }[/math] we can see this set is equivalent to [math] C = \left\{ {\omega |{{\left| {{\omega _ + }} \right|}^2} = {{\left| {{\omega _ - }} \right|}^2}} \right\}[/math].


Now my confusion is on the following, apparently this cone satisfies the following relation : [math] C \cap {S^5} = {S^2} \times {S^2}[/math]

Can someone help me prove this?

>>8004607
To elaborate:
The polynomial ring over an UFD is UFD.
In any ID, prime implies irreducible.¨
In any UFD, irreducbile implies prime.

>>8005161
What the fuck class is this?

>>8005218
That's straight up linear algebra in the picture.

>>8005218
Idk exactly what it should be called.
This is the book we use to give you a good idea.
http://www.springer.com/us/book/9780387986494?token=prtst0416p

>>8005227
Is it? It must be past where I got to with lnear algebra . All I saw was basic matix stuff, the definition of vector spaces and linear transformations, eigenvalue/eigenvectors, rank/nullity theorem, diagonalizability and similar matrices from a 3000 level class on it and basic stuff on direct sums, invariant subspaces, the jordan decomp. theorem, and normed/inner product spaces (not much that was more complicated than Gram-Schmidt as an example) from a 4000 level class on it. I guess this would be in my school's 5000 linear class.

>>8005161

Isn't w \wedge w always 0 for all alternating forms?

Anyway, forgive my ignorace, but if you have a^2+b^2+c^2 = d^2+e^2+f^2
and a^2+b^2+c^2 + d^2+e^2+f^2 = 1,
then a^2+b^2+c^2 = 1/2 and d^2+e^2+f^2 = 1/2. The converse too. So it looks like scaled S^2 x S^2.

>>8005279
Only for k=odd forms. In that problem, [math]\omega[/math] is a 2-form.

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Any recommendations for a baby's first rep theory book?

>>8005284
Oh, I see, thanks. What about the second part? If the components of w+ are a,b,c and of w- are d,e,f, isn't that all this is?

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>>8005350

>>8005350
Etingof
http://www-math.mit.edu/~etingof/replect.pdf

In general what methods should I be good at when solving problems inductively? So far I've came across summations, inequalities and divisibility. Anything else or are those the main methods?

Secondly, where do inductive arguments/proofs come in handy in CS? Where specifically in algorithms analysis does it crop up?

>>7986068
You're calc iii

>>8000000

>>7985727
Cohomology THAT low. I knew this board was for plebs.

>>8005350
Serre's 'Linear Representations of Finite Groups' is good.

>>8000136
No

how do I use group theory to solve a rubiks cube

>>8008152
en.wikipedia.org/wiki/Rubik%27s_Cube_group

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>>7985665
I don't know if this is the right place to ask this, but I'm looking for books to self-teach subjects like logical syntax, algorithm, advanced mathematics.

I didn't like mathematics in high school, but I attribute that more to a bad way of teaching the subject. Now I want to learn these things and I don't know where to start.

I'm looking for things that relate to mathematics, logic, [spoiler]more towards Frege or Carnap. I know /sci/ hates "philosophy", but whatever. [/spoiler]

>>8008240
http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf

>>7985912

F stat for restricted F test is
((RSSr-RSSu)/q)/((RSSu/n-k))
where q is # of restrictions and k is # of parameters in the unrestricted model.

>>7987451
The dimension of M: f(x)=0 depends on the rank of Df.

>>8005350
Serre's Linear Representations of Finite Groups or Prasad's "Representation Theory: A Combinatorial Approach", just really nice.
If you speak French, I would advise Caldero & Germoni's "Histoires hédonistes de groupes et de géométrie" (I'm sure at least the first volume is on libgen). It's not specifically on representation theory but it's a fantastic group theory book, taking the reader from relatively elementary stuff (classification of matrices by rank and similitude, Jordan normal form, Gauss' reduction of quadratic forms) to more sophisticated problems in the second volume (geometry of the grassmannian, quiver representations, algebraic combinatorics) with an emphasis on the links with combinatorics and representation theory.
Plus it's a fun read.

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>>7995595
Shouldn't be using size here, senpai. Not even in quotes.

Turns out every well-order is order-isomorphic to an ordinal. I.e. every well-order can be represented uniquely up to isomorphism. However there are many non-isomorpic ordinals with the same size.

For a set without any additional structure it makes sense to talk about the size or the cardinality. The cardinality of an ordinal A is the least ordinal B such that there is a bijection between A and B.

w, w+1, w+2 has the same cardinality according to Hilbert's hotel argument. However they are non-isomorpic.

>>7995650
So you are asking how the well-ordering on the ordinals loom like? The ordinals are usually defined as pure object, transitive sets. Then a < b if and only if a \subset b.

Can anyone help me understand Cohen's method of forcing? I understand Boolean algebras and modding out ultrafilters, but I don't understand how it all logically fits together into independence proofs. The resources I have skip over a lot of steps to prove important related results as well.

>>8010419
It is very difficult to visualize the whole thing all at once. I have heard people who have been doing forcing at the research level for years say they feel like they still don't truly understand it.

>>8010419
Oh, as >>8010424, I replied before I finished reading your post, and consequently misunderstood.

It sounds like you just have a lousy book. Get a better book, like Kunen's Independence Proofs or Jech's Set Theory.

The latter you should have regardless.

>>8010428
Will do, thanks for the advice.

>>8010439
I recommend the latter, by a significant margin.

Kunen uses the now largely outdated construction in terms of partial orders, while Jech presents the modern and more elegant treatment in terms of Boolean algebras.

>>8009839
>Histoires hédonistes
lel, what does this mean?

>>8010439
Jech also is an indispensable resource for any studying set theorist. It covers all of the fundamental concepts of the field, far more than just forcing.

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>>8010448
The title is basically "hedonistic tales of groups and geometry" (hedonistic = "pleasure seeking")

>>8009839
"Histoires hédonistes de groupes et de géométrie"
Are you positive there is no English translation?

>>7995047
underrated, made me kek

Is there some easy argument to prove
[eqn]\int_A \frac{1}{\sqrt{1- \vert x \vert^2} \ne 0 [/eqn] whre A is the interior of the unit disk? Some kind of tearing/continuity argument with the boundary?

>>8011121
here you go

[math]
\displaystyle \int_A \frac{1}{\sqrt{1- \vert x \vert^2}} \ne 0
[/math]

>>8010419
>Cohen
perhaps this, for a short outlook
https://ncatlab.org/nlab/show/forcing

>>8010975
I'm pretty sure, yeah :/

>>8011121
That function is strictly positive on A

>>8011294
lol thanks

>>7992397
never go full-poly dude

>>7985756
why does the isomorphism have multiple domains?

a(b+c)=ab+ac
Can someone explain this to me?

>>7986080
>>7986086
>>7986224
>Sci helping

Fourier transform prerequisites (Bracewell's textbook, for example)?

What exactly is the nature of your problem?

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Can anyone help me figure out what I'm supposed to be studying?

pic related is my study guide

I'm in differential equations II

>>7985727
Good to know that I know everything before serious business starts.

>>8014546
>Can anyone help me figure out what I'm supposed to be studying?
Hangul

>>8014546
Why is this written partially in Korean?

>>7985727

Why is all of combinatorics put in one single place? There's so many sub-areas in there...

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>>7985665
posted this on /b/
After a lil while some guy figured it out

>>8015017
This thread is for advanced mathematics questions only. No calculus questions allowed.

>>8015017

Is there a nice simple way, or would you just want to do a double integral to solve it?

Like:
[math]\frac{2\int_{0}^{\pi}\int_{0}^{10} (10 - r)r dr\dtheta}{\pi100}[/math]

>>8015053
You don't need anything more than elementary calculus.

Divide it into columns. The probabilistic weight of each column is its height, and each column has an average distance half its height. The semicircle is given by [math] y = \sqrt{100 - x^2} [/math]. Thus the average distance is [math] \displaystyle \int_{-10}^{10} \sqrt{100 - x^2} \cdot \frac{\sqrt{100 - x^2}}{2} dx = \int_{-10}^{10} 50 - \frac{1}{2}x^2 dx$.

Literally a single-variable calculus problem.

>>8015053
You don't need anything more than elementary calculus.

Divide it into columns. The probabilistic weight of each column is its height, and each column has an average distance half its height. The semicircle is given by [math] y = \sqrt{100 - x^2} [/math]. Thus the average distance is [math] \displaystyle \int_{-10}^{10} \sqrt{100 - x^2} \cdot \frac{\sqrt{100 - x^2}}{2} dx = \int_{-10}^{10} 50 - \frac{1}{2}x^2 dx [/math].

Literally a single-variable calculus problem.

>>8015066

I meant a way that didn't involve calculus.

Won't let me delete. Oh well.

>>8015053
Here's how I did it
\[\frac{\int \int_{R} y\cdot dA}{\int \int_{R} 1\cdot dA}\]
sorry if doesn't work correctly

>>8015070

You can "cheat" and use your knowledge of existing volumes.

In three dimensions, picture the volume of revolution of that semicircle around the diameter, a sphere of radius R.

Say D is the expected distance from that axis of a point chosen uniformly at random in the semicircle. Then 2piD is the expected length of the path that this point traces out in the volume of revolution. That means the volume of the sphere is the area of the semicircle multiplied by 2piD. In other words, 4/3 pi R^3 = 1/2 pi R^2 * 2pi D. So D = 4R/(3pi).

>>8015320
yes, that works for centroids

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>>7985665

>>8015624
>math
>hard science

>>7993709

But it seems like a looking at a OTP cipher as something that needs to be "solved" would be in the domain of Chaos Theory. The creator of that picture should have just put "Chaos theory" instead.

>>8014546
Kek

whats [math]\infty * \infty[/math]?

>>8016409
which convolution is that? dirichlet?

>>8016409
Latex neophyte detected. Use \cdot.

In the context of measure theory, [math] \infty \cdot \infty = \infty [/math] by definition. Measure theory tends to operate in the extended real numbers.

Which of the following asymptotic equations are true and why? 10^x - 8^x ~ 9^x and x+1 ~ n + 10^10^10 and arctan(x) ~ pi/2

>>8016478
But so does [math]0 \cdot 0 = 0[/math] and [math]1 \cdot 1 = 1[/math].

I think that's circular reasoning under multiplication.

>>8016854
only the last one is true friend. direct by definition

>>8016890
So what what [math] 0 \cdot 0 = 0 [/math] and [math]1 \cdot 1 = 1 [/math]? [math] \infty \cdot \infty = \infty [/math] is literally the definition of the extension of the multiplication operator to the extended real numbers.

>>8016900
are you sure? These are asymptotic and not regular equations.

>>8016908
>literally the definition of the extension of the multiplication operator to the extended real numbers
That sounds like something I would do, given time.

So the formula for magic square sums predict that 5 would be the line sum of a 2x2 magic square (n (n^2+1))/2. Of course, there isn't a 2x2 magic square, but the average of each direction that is the same (i.e. parallel) is indeed equal to 5. Is this just coincidence, or is there more to how the formula works?