When someone ask us to find the interval of convergence of a series in Calc 1, do that person also ask us to analyze at the extremes of the interval?
I know that analyzing the interval of convergence through, for example, $\lim_{n\to\infty}|a_{n+1}/a_n|<1$ is analyzing whether it is absolutely convergent or not. I think that we are not interested in this
Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks a lot in advance.
Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ?
It is a lot of work to check on associativity imo.
Maybe there is a shortcut around checking associativity ? Or a way to check it faster ?
Is there an easy algoritm to check if a symmetric ratio...
I also know that there is a 4D rep, which is basically the augmentation ideal of the algebra K^5 which KG acts on
let's call them $K$ and $M$
I know from looking at Brauer characters that $M \otimes_{KG} M$ contains the third simple module that I'm looking for, but that would be a 16-dimensional module
either I would need to look at that, or I don't know where to look
I am not working with the group ring at all here. I am saying that if you take the sum of the standard basis vectors, then that gives a copy of the trivial rep.
Hi, does someone know the Grunwald Wang theorem? I got an answer on Mathoverflow which uses it but I don't quite understand how.
As far as I understand, it says that if $K$ is a number field, then any element $x \in K$ which is a square in all but finitely many places of $K$ must be already a square in $K$.
You mean $(\mathbb{Z},+) \to (2\mathbb{Z},+)$, $x \mapsto 2x$ ? It's a group homomorphism because of the distributivity, injective because $2x=2y$ implies $x=y$ and its clearly surjective. So its an isomorphism of groups.
@AkivaWeinberger dysfunctional analysis is just a subclass of nonfunctional analysis. Too many have underestimate just how large any class under the effects of the "non" operator is
Thus one can easily envision in mathematics, dysfunctional analysis is the analysis of pathological functions, spaces and so on, and their properties. In particular, their deviation from nice functions will be an important tool of study
Consider the $(A_k)^{n}$ ,$ k$ is a field an algebraic set is a subset such that there exists a set of polynomials such that for every point on the subset every polynomial gives zero.
now along with the zariski topology i define those sets to be the closed ones.
now i want to identify is a set is algebraic or not
if it is it must be equal to its closure with respect the zariski topology
also every algebraic set has a decomposition of irreducible algebraic proper subsets
now with all that said i think if you are trying to prove a set is not algebraic
these might not be good methods to do so and would be better to just prove there doesnt exist a polynomial such that it has its zeroes on the given set
most of the course was measure theory without the details + a bunch of distributions without explaining why they are important, we only did some order statistics that was actual probability
tbh i still hate working out all the measurability issues u get when u try to study processes but the theory comes out very nice in the end modulo those things
Speaking of measure theory I went through the proof that the axiom of determinacy implies that every subset of $\Bbb R$ is Lebesgue measurable today, very cool stuff
iirc the simplest model for brownian motion is a 1-parameter family of (absolutely) continuous random variables X_t such that for all 0 < t0 < ... < tn < 1, {X_{t_k+1} - X_t}_k are all independent and X_{t+h} - X_t ~ N(0, h)?
Maybe X_t : Omega x I -> R is continuous in the I factor
@manooooh how about something stronger: for every function $f : X \to Y$, the relation $R : X \times X : aRb \iff f(a)=f(b)$ is an equivalence relation
it's essentially the kernel of the function, and you can formulate the first isomorphism theorem using that
@AlessandroCodenotti it's quite interesting that every finite subset is satisfiable... I think I've even seen the result that for every n, ZFC with only Σn replacement is satisfiable
Problem: Let $F_2 = \langle a,b \rangle$ be the free group on two generators. Let $\phi : F_2 \to \Bbb{Z}_2 \oplus \Bbb{Z}_3$ be defined by sending $a$ to $(1 + 2 \Bbb{Z}, 3 \Bbb{Z})$ and $b$ to $(2 \Bbb{Z}, 1 + 3 \Bbb{Z})$. Draw a cover of $S^1 \vee S^1$ whose fundamental group maps isomorphically to $H$ under the homomorphism induced by the covering map.
Okay. I was able to determine that the desired covering space must be a six sheeted cover. When I was speaking with my professor about this, he said that this means you should draw a graph with 6 vertices. Why does six sheeted cover correspond to 6 vertices? What's the reason?
Oh, because the number of sheets the is the cardinality of any fiber. So any point in the space must pull back to 6 different points in 6 disjoint spaces?
i preferred to think of brownian motion as a measure on the space of trajectories bc then i wouldnt have to think about how it looks and i could just prove bullshit about PDEs by appealing to abstract nonsense tbh
and frequently one only cares about things up to the probability law of a process so it's a convenient way to redefine processes if u wanna do PDEs over the space of paths or something where the solution in question should be some probability law
I met her at Mathcamp several years ago. She specializes in braid theory. She won the "Dance your PhD" competition in 2017 (https://youtu.be/MASNukczu5A)
LOL, thanks. Remember that eigenvectors must be nonzero vectors. Any nonzero scalar multiple of an eigenvector is an eigenvector, so you get a whole line (at least), except for the zero vector.
What do you mean by that, @topologicalmagician? What are you sketching? The integral curves (solutions of the differential equation with a given initial condition)?
hi Demonark ... I take it you're still in a decent mood.
@Leaky: I just don't think much about representations or character tables or anything like that. I've taught a little bit of it, but I really don't remember anything useful.
So any group action (everything finite) can be composed into $\bigoplus_{i=1}^n (G/H_i)$ where $H_i$ is a subgroup of $G$ and $G/H_i$ is the left cosets of $H_i$
I have no idea what you're talking about, Leaky. I have a group acting on a set. What are you decomposing? Orbits? So each orbit is the quotient by stabilizer subgroups. OK.
Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks a lot in advance.
Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ?
It is a lot of work to check on associativity imo.
Maybe there is a shortcut around checking associativity ? Or a way to check it faster ?
Is there an easy algoritm to check if a symmetric ratio...
