25 problems

Let $P$ be the poset shown in Figure, and let $G(P)$ be its associated graph. Consider the coefficient-free $R$-system associated with $G(P)$. A strong $\tau$-sequence and the sing…

Let $G_{\beta,n}$ be the support digraph for a discrete family, let $V_n$ be its vertex set, and let $S_n$ be the set of solution states. Write … A phase transition at…

Let $(a_{k,n})$ be the normalized arrival sequence defined by the recurrence described in the source, with $a_k:=1$ as its normalization, and let $a_{k,1}$ denote its value at $1$.…

Let $\mathrm{Col}:\mathbb N\to\mathbb N$ be the Collatz map, and let $\mu$ be a finitely additive $\mathrm{Col}$-invariant measure on $\mathbb N$, with every singleton measurable.…

Maximum-period conjecture. When $\mu\bmod 3=1$, the maximum period is

Characteristic-sequence conjecture. For every transitive $f\in S_n$,

Discrete convex-hull conjecture. There exist $j\in\{1,2,\ldots,k\}$ and $r,s\in I_j$ such that, for some $l\leq k$,

Akiyama–Brunotte–Pethő–Steiner conjecture. Every orbit of $r_{\varphi}$ is periodic.

Let $T_2$ be the graph appearing in the paper, and let a setting be an assignment of edge lengths satisfying the tEoM. The non-constant-solution nonexistence conjecture states that…

Let $T_q$ be the graph under consideration, let $M_q$ be a setting of its edge lengths, and let the tEoM denote the discrete time equations of motion. Assume that all edge lengths…

Stack-sorting growth conjecture. The value of this limit lies in the interval $(1.68,1.73)$.

Extremal degree conjecture. If $T\in H_0(S_n)$ is eventually constant, then

Bubble-sort degree conjecture.

Asymptotic growth conjecture. Any solution to this system grows asymptotically like

Let $q$ be an alphabet size, and let $E[D,q]$ denote the set of expansive networks on a digraph $D$ over an alphabet of size $q$, while $EA[D,q]$ denotes the set of abelian expansi…

Let the dimension $dgeq 1$ be given, and consider binary cellular automata with the von Neumann neighborhood. A cellular automaton is number-conserving when it preserves the total…

Let $z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma-z_n}$ be the complex rational delay difference equation, where $k$ is the delay and $p$ denotes the period of a periodic solution. E…

Let $z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma-z_n}$ be the complex rational delay difference equation, with parameters $\alpha$, $\beta$, $\gamma$ and delay $k$. A period $p$ is…

Let $\mathcal{G}$ be the Lie group of nonlocal symmetry diffeomorphisms associated with the integrable map (respectively, ), and let and denote the corresponding continuous dynamic…

Absence of nontrivial cycles. There do not exist integers $n>2$ and $k>0$ such that $T^k(n)=n$.

Let $T_i(n)$ denote the first time card $i$ is seen for the $n$th time. Novikoff et al.'s bounded-increment conjecture. We have … for all $i$ and $n$. The paper states that this co…

Let … where $\lambda$ is real and $|\lambda|<2$. The map is the discretisation of the linear rotation-like map $(x,y)\mapsto(\lambda x-y,x)$, with $\lambda=2\cos(2\pi\nu)$. Bounded…

Singularity confinement conjecture. Singularity confinement holds generically on $X_S$ unless

Hirota–Kimura integrability conjecture. The Hirota–Kimura discretization remains algebraically completely integrable.

Clebsch discretization conjecture. All claims of the relevant integrability theorems for the special Clebsch discretizations also hold for this discretization of the general Clebsc…