8 problems

Notations and Preliminaries

For integers $m\geq 4$ and $0\leq i\leq m$, define the Boros–Moll numbers by $d_i(m)=2^{-2m}\sum_{k=i}^{m}2^k\binom{2m-2k}{m-k}\binom{m+k}{k}\binom{k}{i}$, and define…

Let each site of $\mathbb{Z}^2$ be assigned an independent uniformly random initial direction among its four outgoing directions. Starting at the origin, the Eulerian walker turns…

The integrable map corresponding to the linear equation, respectively the nonlinear equation, is denoted by … . A nonlocal diffeomorphism is a diffeomorphism whose action is nonloc…

Generalized Collatz convergence and cycle conjecture. For all $p\geq0$ with $p\ne2$, the triplet $(2^{p+1},3\times2^p,2^p)_+$ is strongly admissible of order one, with unique trivi…

Mersenne-triplet admissibility conjecture. For all $p\geq 2$, $(2^{p-1},2^{p}-1,1)_{\pmb{+}}$ is an admissible triplet of order one. It is strongly admissible only for $p=2$ and we…

Let $\mathcal{G}$ be the Lie group of nonlocal symmetry diffeomorphisms associated with the integrable map … . Let its Lie algebra of generators be the Lie algebra of the correspon…

Let a QP differential system be given, and consider the multiplicative and additive discretizations … के and … where each $\varphi_i$ is differentiable. Commutativity conjecture. F…