Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $(1,2)$-GDWN splits   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
$\phi = \frac{1+\sqrt{5}}{2}$ and for all nonnegative integers $n$,
$A_n=\lfloor\phi n \rfloor$ 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 $\alpha \le y/x \le \alpha +\epsilon$, for given real
numbers $\alpha>1$ and $\epsilon > 0$, for which each $(x,y)$ is an N-position,
but that there are infinitely many P-positions for both $1\le y/x <\alpha $ and
$\alpha +\epsilon < 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
$\phi $ 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 $\liminf_{n\rightarrow \infty}\frac{#\{i\mid a_i < n\}}{n} \ge \phi^{-1}$.

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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph