The Long Train Problem

I thought of a problem while waiting at a railroad crossing. Very simply, it goes like this:

When you approach a railroad crossing (with a train crossing it), how many cars do you want to see waiting there?

Assuming, of course, that you want to spend as little time waiting at the intersection as possible, there are arguments to be made for fewer and for more cars. If you see fewer cars, it’s likelier that the total time the train will have spent at the crossing will be small. If there are more cars, it’s likelier that the train has already spent most of its total time at the intersection. So it’s a bit of a paradox, right?

For the sake of this problem, assume that you’re not able to measure how fast or how long the given train is. However, you have two values to work with: R and f(t). R is the number of cars per unit time which approach the crossing, which is assumed to be constant. f(t) is the continuous probability distribution that a train will spend a total time t at the intersection, given that you encountered it at the intersection. (Without that last given condition, the problem would look different, and probably harder.)

I haven’t spent too much time on this problem, but I think it might be a little tricky. If anyone has any thoughts on it, I’d love to hear them!

(Also, let’s say that you consider only the cars on your side of the tracks.)

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Fench article on Akshay Venjatesh, Fields Medal 2018

I’m just a french lover of math and was reading the following article in the french edition of scientific ameican about Akshay Venkatesh and wahoo! (I lack of words)

https://www.pourlascience.fr/sd/mathematiques/akshay-venkatesh-un-mathematicien-batisseur-de-ponts-16726.php

Edit: sorry for typo in title, i’m on mobile

submitted by /u/One_Philosopher
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How do I Project a Fiber of the Hopf Fibration?

Yes, I do know the formula for stereographic projection from R4 to R3.

(a,b,c,d) gets sent to (a/(1-d), b/(1-d), c/(1-d))

What I don’t quite know how to do is project a fiber around S3 (one of its great circles.)

I would assume that we project “sets of points” for each circle by individually “tracing” out each point on the fiber and then projecting each point, forming a villarceau circle in R3.

I do indeed know how to “trace” out a fiber in C2. I have no problem with this. I was having some trouble learning how to trace out a fiber with the versors. I assume, as the wikipedia page has it written, that a fiber around S3 contains the set of quaternion elements where q = u + vp, under the condition that u2 + v2 = 1. Also, in this formula, p is a vector quaternion of the form (0 + bi + cj + dk) with a magnitude of 1 which lies on S2.

So, is that all I need to know in order to trace out a fiber using quaternions? I was reading the wiki page around the section titled, “geometric interpretation using rotations.” They were saying things about p, it’s antipodal point, and 180 degree rotations which I just couldn’t follow. For example, I have no idea how they derived the other formulas such as this one: https://wikimedia.org/api/rest_v1/media/math/render/svg/d60d533f953eed6b8975fd4c6ca39c7f48d41107.

Anyway, when projecting a fiber, I wanted to know if we strictly have to do this using quaternions. Do we take each quaternion of the form (a + bi + cj + dk) and then extract it’s real components, (a,b,c,d) and then project this point, and then repeat this process for every point around the fiber?

Or, do we project from C2 by taking two complex numbers, (z,w), break them up into their components, (a + bi), (c + di) and then extract all of the real numbers (a,b,c,d) and then repeat this process for each point of the fiber?

Or, is there a way to project a fiber directly from C2 to C x R? Instead of the traditional projection from R4 to R3.

submitted by /u/adam717
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The art of problem creation

Let me preface this by saying that I want to become good at posing problems. By good I mean able to produce problems of varying difficulty that are insightful and read well. I understand this is quite general so lets focus on pre-universtiy maths (similar to a-level for those familiar with the uk system) so things like u-substiution, polar coordinates, hyperbolic functions, solving 1st/2nd order DEs.

How do people do it? It seems so difficult to create these seemingly random scenarios and expressions that you want students to prove. Even then, they must not be too challenging and not too easy. For example: https://imgur.com/tuOeaOv

How can I become good at posing such problems?

How “able” should I be mathematically speaking to be able to come up with good problems?

This is all very broad I know, but I just want some guidance. any advice at all is highly appreciated 🙂

submitted by /u/xreputation
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Maths delirium while sleeping

This is gonna sound weird. Over the course of the last few days I have been revising for a school maths exam, pretty much non stop alongside physics. Because of this constant exposure to the equations and numbers I have started to notice myself falling into a brief state of delirium at night. This involves my thoughts about normal everyday things turning into a maths equation in my head. For example I might be thinking about my position in my bed and I will start to treat it as some sort of maths problem with each side of the equation being the side of the bed or some sh*t. Or who I spoke to most in the day will turn into some quadratic inequality. Fortunately I almost instantly return to reality, but I still find this quite odd.

Any thoughts?

submitted by /u/Ikebear2
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Prerequisites for studying the Riemann-Hilbert-Birkhoff Problem.

So I ran into the problem in the title. I want to use it as motivation for studying differential equations, as I’m lacking in this field. I’ve taken some analysis, topology, and algebra courses at the undergrad level. I also took the cookie cutter ODE course intended for all majors. What texts should I read specifically to learn what the problem is even asking. Arnold’s book is one that I’m looking at for a general grasp on the material on a rigorous level.

submitted by /u/Orangefox32
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Do you know any work concerning the length of random walks before they return to their origin?

Specifically, one-dimensional random walks. I’m trying to calculate the average length of random walks before they cross their origin again, like a walk that goes one unit left then one unit right.

Gambler’s Ruin seems to suggest it would be finite, but the sum to find the average almost certainly diverges.

submitted by /u/CoinMarket2
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An interesting sequence

ABAABBAB…

It’s a simple sequence with just 2 elements “A” and “B”, with the minimal number and size of “patterns”. For example, the sequence “AA” can be interpreted as two “A” or “2(A)”, the sequence “ABAB” as “2(AB)” and the sequence “AABB” as “2(A),2(B)”. So, 2(AB), 2(A), 2(ABA) are all patterns with size 2,1 and 3 accordingly. Note that, 2(ABA) represents the sequence “ABAABA” which can also be written as “A,B,2(A),B,A”, but 2(ABA) is the most compact representation and that’s what we’re interested in. The sequence above is a result of the following process:

Choice 1: A → A ✓, B → B ✓ (The first choice doesn’t matter since both sequences can’t have any “patterns”)

[We choose A at random]

Choice 2: AA→2(A) ✖, AB→A,B ✓ (we choose B here since A gives us a “pattern” and B does not)

Choice 3: ABA→A,B,A ✓, ABB→A,2(B) ✖ (now we choose A since B gives us a “pattern” while A does not)

Choice 4: ABAA→A,B,2(A) ✓, ABAB→2(AB) ✖

(choosing A results in a sequence with only 1 pattern of size 1,while B results in a sequence with 1 pattern of size 2, we want to minimize the number and size of these “patterns”, so we choose A)

Choice 5: ABAAA→A,B,3(A) ✖, ABAAB→A,B,2(A),B ✓ (B gives us a smaller pattern)

Between a choice that results in a sequence of 2 patterns of size 1 and another choice which results in a sequence with 1 pattern of size 2, we choose the former. This sequence can be thought of being the least compressible or even the sequence that would appear the most random. It is defined in very loose terms, and that’s kind of the reason why I posted it here.

submitted by /u/jeffnamename
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Self-studying ODE’s, dynamical systems and chaos

I’m a first year maths student and I’d like to do some self-studying of differential equations, dynamical systems and chaos theory, because I can’t fit the courses into my degree, but I’d love to learn.

I was wondering what would be a natural place to start, because I don’t have much background in differential equations. So far in my degree I’ve had introductory courses in Linear Algebra and Analysis. I was thinking of starting with Nonlinear Dynamics and Chaos by Strogatz, but I was wondering if I’ll need to do some extra reading on ODE’s?

submitted by /u/lonely-goth
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