**95**

votes

**1**answer

9k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**70**

votes

**2**answers

105k views

### Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...

**69**

votes

**4**answers

3k views

### Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...

**68**

votes

**0**answers

5k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**63**

votes

**11**answers

12k views

### What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...

**58**

votes

**6**answers

6k views

### Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual ...

**53**

votes

**5**answers

9k views

### The “Dzhanibekov effect” - an exercise in mechanics or fiction? Explain mathematically a video from a space station

The question briefly:
Can one explain the "Dzhanibekov effect" (see youtube videos from space station or comments below) on the basis of the standard rigid body dynamics using Euler's equations? (Or ...

**53**

votes

**4**answers

2k views

### Perron number distribution

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...

**46**

votes

**1**answer

2k views

### A function whose fixed points are the primes

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...

**43**

votes

**5**answers

3k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

**40**

votes

**10**answers

6k views

### The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...

**39**

votes

**3**answers

3k views

### When does iterating z -> z^2 + c have an exact solution?

If one iterates the map z -> z^2 + c there is obviously a simple formula for the sequence one gets if c=0. Less obviously, there is also a simple formula when c = -2 (use the identity 2 cos(2x) = ...

**31**

votes

**2**answers

719 views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**30**

votes

**3**answers

1k views

### How is the Julia set of $fg$ related to the Julia set of $gf$?

Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f ...

**29**

votes

**6**answers

2k views

### Can we actually find any fixed points with Brouwer's theorem?

Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...

**28**

votes

**6**answers

2k views

### Billiard dynamics under gravity

Has the dynamics of billiards in a polygon subject to gravity been
studied?
What I have in mind is something like this:
Still Snell's ...

**28**

votes

**4**answers

2k views

### $\exists$ a shot in ideal pocket billiards?

Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...

**27**

votes

**4**answers

748 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**27**

votes

**3**answers

1k views

### Improving a sequence of 1s and -1s

Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit?
Two examples illustrate what I think ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**26**

votes

**3**answers

2k views

### Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...

**26**

votes

**1**answer

1k views

### solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.)
I saw this amusing derivation ...

**25**

votes

**2**answers

1k views

### Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ ...

**25**

votes

**1**answer

776 views

### A question of ErdÅ‘s on equidistribution

In his book Metric Number Theory, Glyn Harman mentions the following problem he attributes to ErdÅ‘s:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
...

**23**

votes

**3**answers

2k views

### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

**23**

votes

**2**answers

622 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**23**

votes

**3**answers

819 views

### Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...

**22**

votes

**4**answers

1k views

### Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...

**22**

votes

**3**answers

697 views

### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...

**22**

votes

**3**answers

1k views

### square root of diffeomorphism of R: does it always exist?

Let $f:\mathbb R\to\mathbb R$ be a smooth, orientation preserving diffeomorphism of the real line.
Is it the case that there always exist another diffeomorphism $g:\mathbb R\to\mathbb R$
such that ...

**22**

votes

**3**answers

2k views

### Optic fibers after Joseph O'Rourke

Let $\gamma\colon[a,b]\to \mathbb R^3$ be a smooth curve with curvature $< 1$.
Consider a tube, formed by one parameter family of unit circles with center at $\gamma(t)$ in the plane orthogonal to ...

**21**

votes

**3**answers

4k views

### Status of the 196 conjecture?

A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not ...

**21**

votes

**1**answer

1k views

### Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...

**21**

votes

**2**answers

828 views

### Billiard dynamics for multiple balls

I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
...

**20**

votes

**6**answers

1k views

### If you were to axiomatize the notion of entropy …

What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...

**20**

votes

**4**answers

893 views

### Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts just touching the origin pin, and shoots off at angle ...

**19**

votes

**5**answers

811 views

### Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...

**19**

votes

**2**answers

1k views

### How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. ...

**18**

votes

**6**answers

1k views

### Quantitative versions of ergodic theorem

Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...

**18**

votes

**2**answers

701 views

### Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...

**18**

votes

**0**answers

523 views

### Base change for $\sqrt{2}.$

This is a direct follow-up to Conjecture on irrational algebraic numbers.
Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...

**18**

votes

**0**answers

671 views

### Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...

**17**

votes

**4**answers

735 views

### Non-chaotic bouncing-ball curves

I was surprised to learn from two
Mathematica Demos by
Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel
exhibits choatic behavior:
(The PoincarĂ© map ...

**17**

votes

**1**answer

663 views

### Integrability of the Cohen map

In the 1990's, Henri Cohen asked whether the map $(x,y) \mapsto (\sqrt{1+x^2}-y,x)$ from $\mathbb{R}^2$ to itself is integrable. In other words, are the orbits confined to the level curves of some ...

**17**

votes

**1**answer

526 views

### Can the expansion of a large integer in all bases consist of almost all zeroes?

Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...

**17**

votes

**1**answer

977 views

### Generic points and local entropies

Let $X=\{1,\dots,p\}^\mathbb{N}$ be the space of sequences on a finite alphabet with a metric inducing the product topology, and let $\sigma\colon X\to X$ be the shift map. Let $\mu$ be a ...

**16**

votes

**13**answers

3k views

### Is there a “crash-course” book on Abelian varieties (e.g., an introduction for physicists)?

Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...

**16**

votes

**3**answers

669 views

### Counting certain arrangements of n triangles. Does the count grow superexponentially?

Overview
For integers n ≥ 1, let T(n) = {0,1,...,n}n and B(n)= {0,1}n. Note that |T(n)|=(n+1)n and |B(n)| = 2n.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about ...

**16**

votes

**1**answer

476 views

### Growth rate of number of loops in a graph

Let G be a directed graph with a countable number of vertices, and suppose G is strongly connected (given any two vertices v and w, there exists a path from v to w). Fix a base vertex v0∈G, and ...

**15**

votes

**11**answers

3k views

### Importance of Poincaré recurrence theorem? Any example?

Recently I am learning ergodic theory and reading several books about it.
Usually PoincarĂ© recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not ...