16 problems

For $m\in\mathbb{Z}^+$, let $\tau(m)$ be the minimal $n\in\mathbb{N}$ such that $\mathsf{T}^n(m)=1$, when such an $n$ exists, and set $\tau(m)=\infty$ otherwise. Here $\log_2$ deno…

Let $Z=(Z_t)_{t\geq 0}$ be the stable process considered in the paper, let $T$ be the associated stopping time, and let $\alpha$ be its stability index. For every $k>0$, the moment…

Let $\mathcal{L}^1$ denote the class of integrable horizon distributions, and let $\mathcal{G}$ be the class of horizon distributions introduced in the paper; single-threshold algo…

Let a horizon distribution belong to the discrete decreasing hazard-rate class, denoted by $\operatorname{DHR}$, and consider single-threshold stopping algorithms for the random-ho…

For $m\in\mathbb{Z}^+$, let $\tau(m)$ be the minimal $n\in\mathbb{N}$ such that $\mathsf{T}^n(m)=1$, when such an $n$ exists, and set $\tau(m)=\infty$ otherwise. Kontorovich–Lagari…

Let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd positive integer $n$. Large-number Syracuse falling-time conjecture. … for all odd $n\geq 2^{5000}$. This con…

Let $\operatorname{ft}(n)$ denote the falling time associated with the accelerated $3x+1$ map. Large-number falling-time conjecture. … for all $n\geq 2^{500}$. This is a strong qua…

Let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd positive integer $n$. Mersenne Syracuse falling-time conjecture. … for all $\ell\geq 4625$. The assertion agr…

Let $\operatorname{ft}(n)$ denote the falling time associated with the accelerated $3x+1$ map. Mersenne falling-time conjecture. … for all $\ell\geq 133$. The claim is motivated by…

Let $\operatorname{syr}$ be the Syracuse map on odd positive integers, and let $\operatorname{sft}(n)$ denote the Syracuse falling time of an odd integer $n\geq 3$, namely the leas…

The process is a fair game, and consider its maximal gain before the first subsequent loss of a fixed size. The conjecture concerns the expected size of this gain relative to that…

Let $p_n$ denote the sequence of stopping probabilities for the random walk considered in the paper. Limit conjecture. … This is presented as the main question of the problem. The…

Let $p_n$ denote the sequence of stopping probabilities for the random walk considered in the paper. Even-index monotonicity conjecture. … The paper proves the corresponding inequa…

Let $M(t)$ be the expected number of independent uniformly chosen numbers from $[0,1]$ needed until their sum exceeds $t$, and let $N(t)$ be the expected number of independent unif…

Let $B$ be a standard Brownian motion with $B(0)=0$, and for $h>0$ define the first exit time from the interval $[-h,2+h]$ by … The expected exit time is…

For $bbtau(s)=1$, let $z(n)$ denote the quantity defined from the residue-class counts by the paper's stopping-time construction, with $n\geq2$. Power-of-two conjecture. For every…