on an unrelated note, has anyone come across any sort of paper exploring pseudorandom properties of permutations?
in a similar way to pseudorandom graphs and hypergraphs
The reason I ask is: consider the permutation on p-1 elements by sending them each to their inverse mod p
in many ways this family of permutations is far from random, but it apparently has the property that the longest increasing subsequence of this family tends to 2sqrt{p-1}
tbh, i don't know how it works. but, take a look at the definition offered in the link and compare it with what you just said re longest increasing subsequence
I guess something that might be interesting to look into and possibly fruitful for my problem is to study the general theory of pattern avoidance for random permutations
I think I recognize a lot of the mathematics from Dan Romik's book
I guess I'm fundamentally trying to look for something to cling to that implies the 2sqrt{n} upper bound (or lower bound) while being somewhat nonrandom to use as the basis for proving said pattern
on an semi-related note, here's a cute trick you might not have noticed - let X be a random variable corresponding to the number of fixed points of a random permutation
"The free cumulant functional...plays a major role in the theory. It is related to the lattice of noncrossing partitions of the set { 1, ..., n } in the same way in which the classic cumulant functional is related to the lattice of all partitions of that set."
ahah. i was about to suggest that blog post as well :P
@PrinceM I think it's because you're solving a system of polynomial equations $P_i(x_1,x_2,...,x_n) = 0$ and the (vector) solutions $(x_1,x_2,...,x_n)$ need not admit $(0,0,...,0)$ as a solution, i.e. if you have a vector space of solutions it is most likely a vector space without origin, i.e. an affine space.
@PrinceM Affine spaces have a notion of parallelism in it. If you choose an origin, there is not such notion because say any two codimension 1 subspaces intersect at the origin. But if you don't choose an origin, you can "say" that codimension 1 affine subspaces are translates of the vector subspaces. So parallelism works.
This is precisely what happens in $\Bbb A^n$. There is no god given choice of the origin (hence coordinates) in there.
The point is that without a choice of an origin your making parallelism work. That's all.
This is one of the reasons that $\Bbb A^n$ and $k^n$ are set theoretically the same thing yet have different notation, ya know. The latter is a $k$-vector space - that structure is imposed. The former is a geometric object.
$y$ is said to be proportional to $x$ if $y/x$ is a constant. A fixed number; independent of $y$ or $x$. $y/x$ here is $3x$. Very not constant as $x$ varies.
Hey everyone! Does anybody know if there is something like the first isomorphism theorem for groups that applies to differentiable manifolds? I am hoping for some result like : "let F: N-> M be a differentiable map then n1 ~ n2 iff F(n1) = F(n2) is an equivalence relation. If N/~ is a smooth manifold, then F': N/~ -> im(F) is a diffeomorphism"?
@WardBeullens There's an analogue: it's called the universal property of quotient maps in topological spaces. There's nothing special about manifolds in that context.
If I understand i correctly, the universal property says that F':N/~ -> im(F) is continuous and bijective, but can we conclude that F' is a diffeomorphism given the assumtion that N/~ is a differentiable manifold?
@WardBeullens It says that if $f : X \to Y$ is a surjective map and the topology on $Y$ agrees with the final topology with respect to $f$, then given an equivalence relation $\sim$ on $X$ defined by $x \sim x'$ iff $f(x) = f(x')$, the induced map $X/\sim \to Y$ is a homeomorphism.
Yes, I think so. I suppose you also need the quotient map $X \to X/\sim$ to be smooth. You need to impose sufficient conditions so that everything works out.
Fine. If $N/\sim$ has a manifold structure such that the map $N \to N/\sim$ is a submersion (this is, of course, quite rare; more or less this happens when $\sim$ is the quotient by a group acting freely), then you have your theorem, fairly straightforwardly.
Yeah. Results of my finals weren't exceptionally good - 80% is considered more or less average. A bit worried about admission, etc., but it seems I will survive.
Ok, i'll be a bit less general: I have a transitive lie group action from G on M, the stabilizers happen to be the orbits of an action of H on G. I would like to conclude that G/H is diffeomorphic to M. Im quite new to these things, so im sorry if this is really easy
@MikeMiller The reason people get more scores in here is that after we take the 10th grade exams the sheets go to different schools throughout the states for evaluations. But it seems our school got low scores in general, thus there is hope.
@Ward: Ok, I was just asking for your original statement and I couldn't see it. If you have a transitive Lie group action of $G$ on a space $M$, then $M$ is diffeomorphic to $G$ mod the stabilizer. As you say, the proof is quite clean: $G \to M$ factors through $G/H$. Working locally we check 1) the map $G/H$ is smooth 2) It's a submersion. That proves it.
@Balarka See ya. I posted an answer the other day about when a manifold, after you quotient out a submanifold, is still a topological manifold. You might like it.
Of course there are a lot of examples. One can just blowup something at a point and then quotient that exceptional submanifold. But admittedly I am curious about a classification; let me read on.
Yes, but if you don't know how to do this, that's a place to start. Find your favorite reference on how to get smooth structures on the quotients of manifolds by freely acting compact Lie groups, and read that.
@BalarkaSen In any case, there is a full classification of fake $\Bbb{RP}^n$s and $\Bbb{CP}^n$s easily findable. I couldn't find a classification for $\Bbb{HP}^n$ and $\Bbb{OP}^2$. I posted a question about this on MathOverflow; the cases $\Bbb{HP}^2$ and $\Bbb{OP}^2$ have been resolved.
@robjohn hey, did you manage to take a look over my article I sent to you some weeks ago?
@robjohn Based upon that integral I derived a far harder one that I didn't fully finished till now. I know what to do, but there is still a long way to go.
AB is tangent to the circle. Intersection point between AD and point in circle is C. $\frac{AB}{AC} = m$ I need to find the ratio between area of BCD and ABC using just $m$.
I feel that there is not enough data to find it but I'm mistaken. Can someone give me a hint?
Does anyone know of an easier (or less algebraic) way of showing that the matrix of the composition linear maps is the product of the matrices of the linear maps?
Tried proving directly that a CW complex $X$ is contractible if it's the union of an increasing sequence of subcomplexes $X_1\to X_2\to...$ with each inclusion nullhomotopic
I know it can be done with Whitehead
But it's an exercise from my course and we haven't done Whitehead yet. So I spent like 4 hours trying to find a direct proof
Actually since yesterday I can say. I'll email you. I'm also taking an intro to algebra class. It's a freak coincidence that I haven't taken that course in a previous semester.
Need to write my thesis and do a couple of seminar talks too
It's ok, I'm writing an email to you right now actually. I have work to do today (see the imminent email) but right now I'm doing unrelated math because that's fun sometimes.
Ohh, the program is an ordinary European master's degree. I need to get a certain amount of ECTS credits, which I get by taking exams or doing seminars
@TedShifrin Hi. Exams are coming up so I've just been studying. Luckily the content isn't too complex so it's just down to practising questions/learning proofs.
@iwriteonbananas It is not uncommon (I would say ~70% of the time? Probably a bit more) that people go directly to a PhD instead of doing a Masters first, though two of my good friends have masters degrees. A masters program is usually mostly classwork and sometimes has a thesis component. The PhD program at UCLA has a year or so's worth of coursework required (you have to take n classes, more or less whatever you want), and then "go to town on research".
Working problems and understanding examples is always good, @Gridley, although I know some professors expect students basically just to memorize the textbook... :(
If $p\colon E\to B$ is a covering space and $E$ is path connected, true/false that $\pi_1(B)$ acts on $p^{-1}(b_0)$. Of course we all know it's true when $E$ is simply connected.
@TedShifrin I've got one lecturer who actually sets questions (only a couple), based of jokes he has told in his lectures. One example is: Let f(x) = c and g(x) = e^x. The product function fg is afraid of which differential operator?
@TedShifrin Well, the natural thing that acts is the deck transformation group; if $G$ is the subgroup corresponding to the cover, and $N$ its normalizer, this is $\pi_1(B)/N$. So $\pi_1(B)$ acts as $\pi_1(B) \to \pi_1(B)/N \to \text{Sym}(p^{-1}(b_0))$?
@MikeMiller Ah, okay...can't imagine going from bachelor straight to PhD lol. But I think your undergrad thing is a year longer than a typical bachelor. Masters here is pretty much coursework + thesis, too.
I assume he means $d/dx - I$, but where's the humor, GridleyQ? ... At least my students used to know to groan at my horrid puns, but they never showed up on exams.
We also had a question which was to find a parametrisation of a mobius strip (5 marks out of 100) which I thought was slightly unfair in exam conditions (exam only 2 hours long.
OK, @MikeM, that's what I'd originally thought. But one of my two students convinced me he couldn't argue well-definedness without simple connectedness.
@GridleyQ: That's pretty sneaky unless you've seen the idea before.
@TedShifrin The thing you actually want to write down (lift paths) works fine. You don't need simple connectedness for it, unless I'm missing a subtle subtlety. What's his argument?
Yeah, bananas, you're way ahead of most US grad students, except other Europeans and the really gifted kids who went to MIT, Harvard, Chicago, Princeton as undergrads.
Yes, bananas, I thought so. Interestingly, though, I find statements in Munkres and Hatcher about the quotient $\pi_1(B)/p_*\pi_1(E)$ being in bijection with the fiber, and no one says $G$-set.
@MikeM: What he wrote me was: "I'm wanting to use that if alpha is a path from e_0 to e_1 and beta is one from e_1 to e_n, then alpha followed by beta is homotopic to any path from e_0 to e_n" ...
You could also say that there is an "action" of $\Omega B$ on $p^{-1}(b_0)$; this is actually true. You recover the previous statement by passing to homotopy groups.
But $\Omega B$ is a grouplike topological monoid, not a group, so this gives some people incl. me stomachaches.
They both were a bit stymied by my "Reeb foliation" leaf space question. I took the lines $x=\pm\pi/2$ and the level sets of $y-\sec x$ and asked what the leaf space was. They both disagree with me.
@iwriteonbananas If you want to work without modding out what I was saying before still applies: you have a monoid homomorphism $\Omega B \to \text{Aut}(p^{-1}(b_0))$.
Right, that makes sense. I think I saw that in May's book. Good point though, if we pass to the fundamental group we need to apply $\pi_0$ on the right
Which of Hatcher's notes do you mean? The bit on fibrations in his book, or are there some other notes he put out?
@TedShifrin est ce que $f(t)=\sqrt{t}$ est un bon contre exemple pour montrer qur l'inégalité $\int_0^1 |t| |f(t)| dt \leq \int_0^1 |t| dt \times \int_0^1 |f(t)| dt$
With both you and bananas here, maybe I will advertise a question I answered which I had a great time with. It's not completely answered; I asked an MO question about the last detail.
I would also like to advertise studiousus's incredible answer here.
I don't have time to read those carefully now. Studiosus is impressive. And I'm glad you're getting to know Andy Hwang. He's a good friend and enjoyable mathematician.
@MikeMiller Emailed you. I'm not sure how exactly it works there. Whether or not I need to choose a focus or whatever. But I don't think so. There's a ton of really cool sounding lectures that I want to do, let me see if I can find a list
@MikeMiller Asaf Schachar always asks good question, I'll have a look at your answer
I think it's more that people usually tend to start choosing things to pay more attention on, doing coursework that will prepare them for research in some area
I think the Guillemin & Pollack course my 3rd year of undergraduate is what aimed me toward geometry ... and then a complex manifolds course my 4th year sealed it.
@iwriteonbananas 1) Thanks for the email, I found it helpful. I am on my phone so will probably not respond now, but will later. We should correspond more.
2) I just looked at the faculty list in topology at your upcoming university. I think you'll be very happy there.
@iwriteonbananas You've expressed a taste in both geometric and (sometimes computationally grounded) algebraic topology. Your new place has some of the world's best people at using algebraic topology for geometric purposes.
