1,761 problems

For every positive integer $n\in\mathbb{Z}_{>0}$, define the Collatz map $T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ by $T(n)=n/2$ if $n$ is even and $T(n)=3n+1$ if $n$ is odd. The conje…

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$. The system is uniformly ex…

Let $D(3,n)$ denote the dynamical system governing the relevant tripod Nim arrays, and consider its periodic orbits. Period-divisor conjecture. Every periodic orbit of $D(3,n)$ has…

Manifold decomposition conjecture. For every graph, the set of equilibria of the Kuramoto network is a finite union of manifolds. The question arises because the equilibrium set is…

Let $G$ denote Moroz's map from modified Engel expansions to Pierce expansions, associated with the digit correspondence $p_n=e_n+n-2$, where $(e_n)$ is the Engel digit sequence an…

Let $T:[0,1]\to\mathbb{R}$ be the classical Takagi function, $T(x)=\sum_{k=0}^{\infty}2^{-k}\operatorname{dist}(2^k x,\mathbb{Z})$. For $y\in[0,\tfrac{2}{3}]$, define…

For every choice of positive masses $m_1,m_2,m_3,m_4>0$ and every prescribed cyclic ordering of four bodies, there exists exactly one strictly convex planar four-body central confi…

For the SIRS epidemic model with nonlinear incidence rate $\frac{kI^p}{1+\alpha I^q}$, where $p>0$ and $q\ge 0$ are arbitrary real numbers, determine the largest possible codimensi…

Let a mass-action reaction network with stoichiometric subspace $S\subseteq\mathbb{R}^n$ be complex balanced, and let $\dot{x}=f(x)$ be its mass-action system. For every initial co…

For every smooth magnetohydrostatic equilibrium $(\Omega,p,B)$ in Euclidean $3$-space, where $\Omega$ is an embedded solid torus, the regular level sets of the pressure $p$ are nes…

Let $\Gamma<\mathrm{PSL}_2(\mathbb{R})$ be a cocompact Fuchsian group, and let $\mu$ be a finitely supported non-degenerate probability measure on $\Gamma$. If $\nu_\mu$ denotes th…

Let $f$ be an analytic map defined near a fixed point $p$, with $f(p)=p$. Assume that the linearization $A=Df(p)$ is semisimple and fully stable, meaning that every eigenvalue of…

Let $d\ge 1$ and let $\Phi=\{\phi_i\}_{i=1}^m$ be a finite iterated function system of similitudes on $\mathbb{R}^d$ with non-singleton attractor $K$. Suppose that $\Phi$ satisfies…

For every finitely generated commutative monoid $M$ and every bounded-to-one Borel action $M\curvearrowright X$, the action has finite Borel asymptotic dimension:…

Let $M$ be a complete surface and let $f\colon M\to M$ be a $C^{1}$ diffeomorphism satisfying Condition A: $f$ has dense periodic points, there is a $Df$-invariant splitting…

For every nondegenerate tetrahedron $T_0\subset\mathbb{R}^3$ and every resolution of ties between equally long edges, repeatedly bisecting a longest edge produces a family…

Does there exist an integer $k\ge 1$ and a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields $\mathbf u\colon\mathbb T^3\to\mathbb R^3$ such that ev…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with real-analytic boundary containing the segment joining the two centres, and let $B_h$ denote the billiard map on the fixed-e…

For every free minimal $\mathbb{Z}^d$-odometer $T$ on a Cantor space and every minimal bounded speedup $S$ of $T$ (in particular, every minimal constant speedup), the actions $S$ a…

For every punctured surface $\Sigma_{g,p}$ with $g\geq 2$, the mapping class group $\operatorname{Mod}(\Sigma_{g,p})$ acts ergodically on every non-Teichmüller component of the rel…

For every integer $n\ge 4$ and every choice of positive masses $m_1,\ldots,m_n$, consider the Newtonian $n$-body problem in $\mathbb{R}^3$. Let $\mathcal{S}_n$ be the set of collis…

For every dynamical system $(X,T)$ on a compact metric space with zero topological entropy, every $f\in C(X)$, and every $x\in X$, one has…

For every pair $(x,y)\in\mathbb{R}^{2}$, where $\langle t\rangle$ denotes the distance from $t$ to the nearest integer, one has…

Let $K$ be a finitely generated field, let $X$ be a quasi-projective variety over $K$, and let $\overline{L}$ be an adelically metrized line bundle satisfying the hypotheses of Yua…

For every $\varepsilon>0$, there exists a finite simple graph $G$ on $n$ vertices such that $\delta(G)\geq (3/4-\varepsilon)n$ and $G$ is not globally synchronizing for the Kuramot…