$(1,2)$ - GDWN拆分
$(1,2)$-GDWN splits
摘要：
我们研究了2个无序堆的令牌$(x,y)$的有限非负数的公平拿走游戏。两个玩家根据一定的规则交替从各自的堆中取出至少一个和最多所有的代币，并且当玩家反过来无法移动时游戏终止。我们遵循正常的游戏规则，这意味着不能移动的玩家输了。在Wythoff Nim游戏中，玩家可以从其中一堆中取出任意数量的代币，或者从两堆中取出相同数量的代币。令$ϕ= \frac{1+\sqrt{5}}{2}$和对于所有非负整数$n$，$A_n=\lfloorϕn \rfloor$和$B_n=A_n+n$。Wythoff Nim的P-位置是分别具有$A_n$和$B_n$令牌的所有桩对。我们研究了一个概括的这个游戏称为PDMATHAMT，其中，除了Wythoff尼姆的规则，玩家可以选择从其中一个桩中删除一个积极的令牌数量，并从另一个桩中删除该数量的两倍。我们证明了对于给定的实数$α>1$和$ε> 0$，存在一个无穷扇区$α\le y/x \le α+ε$，其中每个$(x,y)$是一个N-位置，但是对于$1\le y/x <α$和$α+ε< y/x $，存在无穷多个P-位置.这证明了最近一篇论文中的一个猜想。也就是说，邻接的集合在$(1,2)\G$ \emph{splits}中移动Wythoff Nim的斜率$ϕ$ P-位置的梁。我们还提供了一个下界上的较低的渐近密度较低的堆高度的P-位置的Wythoff尼姆的扩展。设$(a_i)$和$(b_i)$，$i>0$是自然数上的一对所谓互补序列，满足$(a_i)$递增，对所有$i$，$a_i<b_i$，对所有$i\ne j$，$b_i-a_i\ne b_j-a_j$ .然后是$\liminf_{n\rightarrow \infty}\frac{#\{i\mid a_i < n\}}{n} \ge ϕ^{-1}$。
英文摘要：
We study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens $(x,y)$. Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the game terminates when a player in turn is unable to move. We follow the normal play convention, which means that a player who cannot move loses. In the game of Wythoff Nim, a player is allowed to remove either any number of tokens from precisely one of the piles or the same number of tokens from both. Let $ϕ= \frac{1+\sqrt{5}}{2}$ and for all nonnegative integers $n$, $A_n=\lfloorϕn \rfloor$ and $B_n=A_n+n$. The P-positions of Wythoff Nim are all pairs of piles with $A_n$ and $B_n$ tokens respectively. We study a generalization of this game called $(1,2)\G$ where, in addition to the rules of Wythoff Nim, a player has the choice to remove a positive number of tokens from one of the piles and twice that number from the other pile. We show that there is an infinite sector $α\le y/x \le α+ε$, for given real numbers $α>1$ and $ε> 0$, for which each $(x,y)$ is an N-position, but that there are infinitely many P-positions for both $1\le y/x <α$ and $α+ε< y/x $. This proves a conjecture from a recent paper. Namely, the adjoined set of moves in $(1,2)\G$ \emph{splits} the beam of slope $ϕ$ P-positions of Wythoff Nim. We also provide a lower bound on the lower asymtotic density of lower pile heights of P-positions for extensions of Wythoff Nim. Suppose that $(a_i)$ and $(b_i)$, $i>0$, is a pair of so-called complementary sequences on the natural numbers which satisfy $(a_i)$ is increasing and for all $i$, $a_i<b_i$, for all $i\ne j$, $b_i-a_i\ne b_j-a_j$. Then $\liminf_{n\rightarrow \infty}\frac{#\{i\mid a_i < n\}}{n} \ge ϕ^{-1}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
