/mg/ MATH GENERAL

pre med student here, aspiring psychiatrist specifically.

i need to focus on doing some self study on statistics during the summer. could someone recommend a good online resource(s) to do such a thing?

pic completely unrelated as it only exists to get more attention. she goes by 'maserati xxx' in case you're wondering.

right now I'm studying for my game theory exam... my interest is differential topology though, I'm writing my thesis on surgery theory, so Milnor's H-cobordism as the main literature together with Hirsch, Brocker & Janich and Kosinski as supplements. Also Guillemin & Golubitsky's book on stability theory is a VERY good introduction into function spaces of manifolds, something which I absolutely couldn't digest from Hirsch. Damn, so many good books, I'll have a busy summer.

Noncommutative Geometry

What books are you reading?

Good suggestion for function spaces of manifolds!

>What books are you reading?
I don't know of any books on the topic. I am currently reading this arxiv.org/pdf/1402.7364v2.pdf

zoophilia belongs to /b/

khanacademy.org/math/statistics-probability

This will do everything you need. Probably more.

thanks so much, i'll start here.

pic unrelated again, but the source is 'bria myles' in case anyone wants more thicc chocolate.

read godel

You were already told on lit to read Brand and Witty.

I've gotta choose a master's thesis, and im torn between

>Differential topology (reference books: Milnor topology from a differentiable viewpoint, Bredon Topology and geometry, Differential topology Hirsh)

>manifolds, vector bundles, de Rham cohomology, framed cobordism. Describe exotic spheres with connections to number theory.

OR

>Topology of Vector Fields and Vector Bundles (reference books: Milnor, characteristic classes, Hatcher Algebraic topology, Hatcher Vector Bundles and K-Theory

>poincare-hopf theorem, parallelisable manifolds, tangent bundles of manifolds, vector bundles over spaces, characteristic classes

I want to end up doing a PhD after my masters (although I'm also interested in Algebraic geometry)

Can anyone help me with the following problem?

The problem is to find the residue field of each point of [math]\operatorname{Spec} {\mathbb{F}_p}\left[ x \right][/math] and the count how many points per residue field.

I know the points are and where f is any monic irreducible polynomial over [math]{\mathbb{F}_p}[/math]. I also know that if f is of degree n then the residue field will be [math]{\mathbb{F}_{{p^n}}}[/math]. So pretty much I just want to know how to count the number of monic irreducible polynomials of degree n for each n.

...

math.stackexchange.com/questions/152880/how-many-irreducible-polynomials-of-degree-n-exist-over-mathbbf-p

Denote by [math]\Pi_n[/math] the set of monic irreducible polynomials of degree [math]n[/math] of [math]\mathbb F_p[t][/math].
Notice that, for each [math]n \ge 1[/math], [eqn]t^{p^n}-t = \prod_{\beta \in \mathbb F_{p^n}} (t-\beta) = \prod_{d\, |\, n}\prod_{P \in \Pi_d} P[/eqn]
Indeed, the degree of each element in [math]\mathbb F_{p^n}[/math] divides [math]n[/math]. Conversely, if [math]d |n[/math], and [math]P \in \Pi_d[/math], then it splits in [math]\mathbb F_{p^n}[/math] and thus divides [math]t^{p^n}-t[/math]. Since [math]t^{p^n}-t[/math] and its derivative are coprime, we get the desired equality.
Now, taking the degrees on each side, we get [eqn]p^n = \sum_{d|n} d|\Pi_d|[/eqn]
Using Möbius inversion, we get: [eqn]|\Pi_n| = \frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)p^d[/eqn]

>finish masters degree
>get offered PhD position
>ask the prof I'm going to work with for some project ideas
>tells me to read some articles on arXiv and recreate the data
>do so within 5 months
>email her about it
>"oh shit lmao lemme get back to you on that sonic the hedgehog"
>email her colleague I want to work with for some extra work
>same thing happens
>so bored that I'm smoking 2 to 3 cigarettes a day
>PhD doesn't start for another 3 months

Find a different Ph.D. position.

Do symplectic geometry instead.

fug mayne, which one is more interesting/rich/more useful to know?

> for some time been haunted by the feeling that mathematics is just a symbol-game.

That's what you get for studying anything else than analysis.

Veeky Forums here. I've been trying to find a way to calculate the odds of rolling certain outcomes of a resolution systems, but I just can't do it and neither can anyone else judging from the fact that I haven't found any tables anything proper at all. So, you are my last hope Veeky Forums

What I want to do is find a way to calculate the probability to roll a set of any number on a certain number of dice with a certain amount of faces. For example
>Three of any kind on 7 six sided dice
>Two of any kind on 5 ten sided dice

I figured I'd do this with good, old binomial distribution, figuring
>n = number of dice
>k = number of dice showing the same face
>p = 1/number of faces on a single die

And go from there. If I wanted to know the probability of rolling three tens, that'd be easy, but somehow I can't manage to change the thing so I can get the odds of rolling three of any number.

>If I wanted to know the probability of rolling three tens, that'd be easy, but somehow I can't manage to change the thing so I can get the odds of rolling three of any number.
Can't you just multiply by the number of sides of the die? You'll have the exact same chance of rolling 3 10s as you will of rolling 3 of _any_ number

>calculating all the die results as though they are dependant

Time to put yourself down kiddo.

That gives me 124% chance to roll doubles on 7 ten sided dice, which obviously isn't right

Don't do both, and do not do symplectic geometry unless it really interests you. It is a common pidgeon hole.

After searching a bit on Google I've found this link math.stackexchange.com/questions/1042367/probability-of-rolling-doubles-from-5-dice
which suggests to apply this equation for finding the probability of obtaining at least a double with 7 ten sided dice:
[eqn] P(X \ge 2) = 1 - \frac{\prod_{i = 0} ^ {7-1} 10 - i}{10^7}[/eqn]
which basically means that you find the probability of not obtaining doubles and you subtract it from the total. Unfortunately I'm a finance student, so I can't help you too much with this kind of things, you can only hope that someone who studied combinatorics will give you a more sophisticated formula.

What I wanted to ask to the mathematicians here is: did you find the problems on Project Euler difficult? I started to do the first fifty easiest ones, but I've only managed to complete 27 of them. I find that some of them are so strange that I can't even fathom how I can tackle the problem and I am wondering if I am that stupid or even other people have found some difficulties (the fact that I'm stupid is obvious, I just want to know how much I am).

How are you finding Aluffi's?

The distribution is not binomial because there is independence in the dice rolls. The outcome of one roll does not lend any information as to what the outcome of the second, third, etc. roll is.

Because you have independence, you can calculate the probability of any specific set of dice rolls by

[(1/n)^(k-m)*(n-1)/n^k].

Where n is the number of sides, k is the number of dice, and m is the number of arbitrarily defined values you want.

I fucked up the equation initially,

P = [(1/n)^(m)*(n-1)/n^(k-m)].

That should be correct.

How do (or did) you guys study undergrad math? I feel like I'm spending too long doing questions, moving at to slow a pace.
Also any study tips in general?

Imagine you are an undergraduate student. You have 5 courses a semester (4 months long), an assignment in each course due every week, and you have to finish the course by the end of the 4 months. At two months, you have a midterm exam, and after the 4 months, you have a final.

Now if you are doing self-study, you are going to have to imagine doing these things, but the way you want to. I plan the book, splitting it off into weeks (according to the 4 month rule) and then I take interesting questions from each of the weekly sections that will constitute an assignment. I do this weekly for each of my 5 selected topics. At half way point, I pick up the book and select exercises I never got to do that are still interesting to me from the weekly sections prior to the two months mark, and then I do those as a sort of midterm. The final exam is similar, just for the whole book itself.

This way, I get to study what I want, when I want, and I also get some sort of rigour, as well as planning the course myself.

Move slowly, read the question/theorem carefully, and try to appreciate the quality of the meaning before moving on. If you do this enough you will begin to develop experience in mathematical understanding and problem solving which will make you faster at solving problems in the future.

I usually ask myself the question, what does that mean when studying mathematical content. If I can't answer the question what does that mean, then I do not understand how it works but I could understand how to use it.

Studying takes time.

> (although I'm also interested in Algebraic geometry)

Then do complex geometry. You can switch back and forth between analytic and algebraic techniques at will.

Wait, a binomial works precisely because the events are independent, it's even in the first paragraph of wikipedia en.wikipedia.org/wiki/Binomial_distribution
The problem is that if you multiply the probability of obtaining a double for the number of sides of a die you repeat the results where there are two or more doubles, but if you can manage to subtract those result a binomial should be applicable.

These graphs are not isomorphic*, right? Cause my retarded text book says they are.

*The left one has a vertex of degree 2 that it is adjacent to exactly two vertices and both of them are of degree 3. The right one has no such vertex.

yeah, but which one of those two masters options gives me a better direction towards AG? I'm assuming the second, but the first one looks pretty tasty ngl

I'm working with a friend to solve the collatz conjecture, I think we're decently close to a solutions

one is cis, the other is trans, senpai

Second.

Vector Bundles and K-Theory

One of my favorite books so far. I appreciate the categorical viewpoint (granted I had some undergraduate algebra knowledge before).

aluffi is supposed to be graduate anyways

I'm not a graduate though. This doesn't really matter, it's more about how much math you've already done and how familiar with proofs in algebra you are. Anyway Aluffi is good, I learnt basic algebra with Artin/Herstein but this is quite different.

>more interesting
Symplectic geometry because it's how quantization in physics is done.
>rich
Vector bundles since you're going to mix algebra with topology when you study (co)homology.
>useful
??? None of them lmao try engineering instead.

>still no proof of the Riemann hypothesis

>Decries the non existence of a proof
>Doesn't prove it himself

Enjoy your never-ending inferiority, brainlet.

everyone who wanted to prove it (and is not a brainlet), has already proved it and moved on. The task of writing it up is left for the first brainlet that thinks he discovered it.

>First semester
>Just read a proof for why Euler's totient function is multiplicative

I didn't understand it initially, only after reading the proof a 2nd time and very slowly. I'm really far behind in my Linear Algebra course. I'm starting to think, that I'm a brainlet, what do?

how interesting is Abstract Algebra? as an applied math major, it's not required.

what kind of problems advanced geomtry (Algebraic/simplectic/differential) tries to solve?
and what are categories all about?

So basically the problem is that I'm calculating the probability of 2+ dice showing the same face and not EXACTLY two dice showing the same face?
I'm really having trouble to wrap my head around this stuff, which is why I've ended up coming here after a few weeks of trying to solve and research this myself.

Algebra finds itself in multiple applied math fields though, cryptography being one of the most obvious. CS comes to mind too.

how to solve polynomial equations, mostly

i'm getting my ms in applied math right now. if i continue on in science i'm going back to physics for my phd. math is brutal and rigorous and i find physics a more creative outlet for my brain.

Algebraic geometry does pretty much, but does have other purposes.

Symplectic Geometry I am not so sure about. I believe it originates from Hamiltonian Mechanics and thus has various purposes in Classical Mechanics, ODEs, Dynamics, etc..

Differential Geometry was originally just about studying invariants of smooth curves and surfaces, grew into higher dimensional manifolds. But more modern work has Differential Geometry intertwining with the study of PDEs, although the original motivations still remain at times.

All 3 subjects overlap in the study of Kahler Manifolds.

Categories can be used for various purposes. Could just be to classify a certain type of mathematical object, like take a category of Groups or a category of Topological Spaces, etc.

They could also be used to provide the basis of mathematical objects themselves (like how a set usually underlies an object), ex. Sites, Stacks, etc.

Sometimes you can associate certain categories to specific objects and the category will contain certain information about said object. ex. Derived Category of a Scheme, Fukaya Category of a Symplectic Manifold, etc.

thanks.
hwhats a khaler manifold?

A Kahler manifold is a complex manifold, with a certain type of metric (Hermitian metric that is flat up to second order) called a Kahler metric. The Kahler metric defines a symplectic form on the manifold.

When a Kahler Manifold is compact and the cohomology class of its symplectic form is rational, then it is equivalent to a smooth projective variety.

It's fucking up because you are asking for different pieces of information and no one general formula can provide both.
The probability of a set of die rolling any given result is a very different question to the probability of a unique string of results appearing within your rolls.

Symplectic geometry is differential geometry of symplectic manifolds: a manifold with a symplectic form, something akin to an inner product, except this time alternating (=0). A symplectic form is to volume as an inner product is to length, roughly.

By necessity, symplectic manifolds are even dimensional, there is an odd counterpart, contact geometry. Kahler geometry falls more in the realm of symplectic geometry than in algebraic geometry by far.

Differential geometry is a very catch-all term for the study of manifolds, and hence includes symplectic geometry as a subset.

Algebraic geometry studies what are called algebraic varieties, the set of common solutions to a family of polynomials. You can think of things like cones and spheres as varieties, as well as conic sections. Algebraic geometry varies from classical where it is very much focussed on surfaces and curves defined by polynomial equations, all the way to the more modern interpretation which uses schemes, a generalization of a variety which heavily relies on category theory to provide an easier language to discuss them in.

Category theory is an alternative way of looking at mathematics, rather than through the lens of set theory. Categories have two parts to it: a collection of objects, and a collection of "morphisms" or "arrows" between objects, all these things obeying a few axioms such as composition and existence of identity morphisms. The language of categories is very useful for generalizations and abstractions which leads to more profound connections between subjects.

does category theory replicate math the same way as it is done in set theory?

>Kahler geometry falls more in the realm of symplectic geometry than in algebraic geometry by far.

Well when the Kahler manifold is nice enough (i.e. compact w/ rational Kahler class) you can switch back and forth between studying it from algebraic and symplectic perspectives. In very special cases, those perspectives might even be equivalent.

Set theory has merits for being easy to understand and even finds itself embedded in category theory as a tool for explanation. On the other hand, category theory cannot really replace set theory in the way we might wish it would, because of how much abstraction there is. It is difficult to do math with pure category theory. Category theory has certain places it is useful and not useful. It is particularly useful for explaining structures that are a kin to the object/morphism structure.

No doubt that algebraic geometry has applications in Kahler geometry, but I don't think that is enough to say that it is the intersection of algebraic and symplectic geometry. A lot of it could care less about algebraic geometry.

>but I don't think that is enough to say that it is the intersection of algebraic and symplectic geometry

It is considering they are the only spaces that can be both a symplectic manifold and a projective variety.

I hate to be pedantic, but there exist Kahler manifolds which are not varieties. Of course, then they are not Moishezon, but that's okay---still Kahler.

>I hate to be pedantic, but there exist Kahler manifolds which are not varieties

I know. I said earlier they must be compact w/ rational Kahler class (as per Kodaira). Obviously not every scheme is a Kahler manifold either, thats why I said their intersection.

And thus algebraic geometry intersects with Kahler manifolds, but does not encompass it, as the statement "kahler geometry is the intersection of AG and SG" would imply. That's all I am saying.

But isn't there a way to get from "Exactly two in seven dice rolling tens" to "Exactly two in seven dice rolling any of the same number"?

Maybe you could try to write a script for checking all the possible combinations by brute force and then you remove the instances where there are more than one double, but it's a huge hassle honestly. (if you google a bit you can easily find some programs which produce a matrix of all the combinations, but it's the part of checking if in a combinations there's more than one double that I find complicated)

thanks, very clear explanation user

Ask Stackexchange, please. No one here knows, it seems.

I've considered that and by now it'd probably be less of a hassle, but I'm geniuinely curious by now hence why I'm sticking to it.

I guess I'll do that in the end. I had another go at it, so I'll toss that one out here and see if it leads somewhere, though.
I thought of it more in the way of combinatorics:
There are 7c2 ways to pick the two dice that show the same face and 10c1 possibilities for the face. For the remaining 5 dice, there are consequently 9c5 possible combinations of different faces if I don't want any other pairs, triples quads or quints. So basically

[math] {10*{{n}\choose{k}}*{{9}\choose{n-k}}}/10^{10}}[/math]

[spoiler]I hope this works, or I'm going to look very silly[/spoiler]

I feel like this might be correct, but I also might be missing something.

Well, I fucked that one up. Good job.

nice meme, any chance you'd be willing to tell more

Wittgenstein did a whole lot of writing on language-games, which are somewhat analogous to what you mention. However, from a philosophical point of view he cannot be considered entry-level.

If anyone were close to solving , then they would not be bragging about it on Veeky Forums.

bump

Fix [math]k\in \mathbb{N}[/math] and let [math]\lambda\in P^{+}(k)[/math] be the set of initegrable highest weights of level [math]k[/math] of the representatoins [math]V_\lambda[/math] of the affine Lie algebra [math]\hat{\mathfrak{g}}[/math]. Let [math]L \subset \mathbb{R}^3[/math] be an oriented framed link and let [math]t_j, 1\leq j \leq n[/math] be straight lines that decompose [math]L[/math] into elementary tangles up to isotopy. In each elementary tangle, we assign the highest weights [math]\lambda_j ~(\lambda_j^*) \in P^+(k), 1\leq j \leq m[/math] to each intersection [math]q_j[/math] of the link [math]L[/math] with [math]t_j[/math] if the link segment is oriented downward (upward). Set [math]V(t_j) = V_{\lambda_1\dots\lambda_m}[/math] to be the space of conformal blocks at [math]t = t_j[/math] with weight [math]\lambda_j[/math].
Define a linear map [math]Z_j:V(t_j) \rightarrow V(t_{j+1})[/math] and let [math] Z(L,\lambda_1,\dots,\lambda_m) = (Z_{n}\circ \dots \circ Z_1)(1)[/math], then [math]J(L,\lambda_1,\dots,\lambda_m) = \prod_{j=1}^{m}Z(K_0;\lambda_j)^{-\mu(j)} Z(L,\lambda_1,\dots,\lambda_m)[/math] is a link invariant, where [math]K_0[/math] is the [math](0,0)[/math] link with two minimal and two maximal points and [math]\mu(j)[/math] is the number of maximal points in the component [math]L_j[/math] of the link [math]L[/math].
With [math]J[/math] and Dehn surgery on 3-manifolds [math]M[/math], we can now define the Witten invariant by [eqn]Z_k(M) = S_{00}e^{-\pi i \sigma(L)\frac{c}{4}}\sum_\lambda S_{0\lambda_1}\dots S_{0\lambda_m}J(;\lambda_1,\dots,\lambda_m)[/eqn], where [math]S_{\mu\nu}[/math] is the representation of the action of the conformal group on level [math]k[/math] characters [math]\chi_\lambda[/math] of [math]\mathfrak{g}[/math], and [math]\sigma(L) = \#\{n \in \operatorname{Spec}(L_i \cdotp L_j) \mid n > 0\} - \#\{n \in \operatorname{Spec}(L_i \cdotp L_j) \mid n < 0\}[/math] is the signature of the link [math]L[/math].

Complex

Prove the argument principle.

Riemann mapping theorem is cool. Never used it tho. Dont know if i ever will

what's the math behind a multiplayer server/client

i know about lamport timestamps which are a way to keep track of what order asynchronous data are supposed to be in, but it's not that simple, is it?

imagine living in that 80 year period where no one knew the answer to the continuum hypothesis, must have been infuriating

Probably a handful of temporal logic

this is kind of pointless, but the number of square-free numbers less than x equals

x*6/pi^2 - sum(mobius(n)*frac(x/n^2)

where the sum is from n=1 to n=infinity

I think these two papers are fairly representative of the work I've been doing.

>On The Riemann Zeta Function
>vixra.org/abs/1703.0073
Here I make an argument against the Riemann hypothesis.

>Quantum Gravity
>vixra.org/abs/1506.0055
Here I show how the framework I use to argue against Riemann hypothesis is also able to produce the dimensionless constants of gravitation and quantum theory: [math]8\pi[/math] from Einstein's equation and the fine structure constant of quantum theory.

but everybody knows the Riemann hypothesis is true.

>linking a document with your actual name on Veeky Forums

>it is the universe's ultimate symbol game since it has been proven to be so useful for scientific models.
but what scientists do has nothing to do with the secrets of the universe

How hard is Parseval's theorem to prove for a sophomore?

Instead of telling us your year you should tell us how much you know.

>two column papers

Uniform convergence, normal convergence, asymptotical summation, power series, d'Alembert's criterion, Abel transformation, Minkowsky's inequality, Lebesgue-Riemann lemma, real-valued inner product spaces (not Hilbert spaces), kernel decomposition, group theory 101, discrete probability theory.