Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Generalized Diagonal Wythoff Nim   In this paper we study a family of 2-pile Take Away games, that we denote by
Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose
sets of options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and
$\{(t,t)\mid t\in \M \}$ respectively. If we to 2-pile Nim adjoin the
main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as options, the new game is
Wythoff Nim. It is well-known that the $P$-positions of this game lie on two
'beams' originating at the origin with slopes $\Phi= \frac{1+\sqrt{5}}{2}>1$
and $\frac{1}{\Phi} < 1$. Hence one may think of this as if, in the process of
going from Nim to Wythoff Nim, the set of $P$-positions has \emph{split} and
landed some distance off the main diagonal. This geometrical observation has
motivated us to ask the following intuitive question. Does this splitting of
the set of $P$-positions continue in some meaningful way if we, to the game of
Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move
of the form $\{pt, qt\}$, where $0 < p < q$ are fixed positive integers and $t
> 0$? Does the answer perhaps depend on the specific values of $p$ and $q$? We
state three conjectures of which the weakest form is: $\lim_{t\in
\N}\frac{b_t}{a_t}$ exists, and equals $\Phi$, if and only if $(p, q)$ is a
certain \emph{non-splitting pair}, and where $\{\{a_t, b_t\}\}$ represents the
set of $P$-positions of the new game. Then we prove this conjecture for the
special case $(p,q) = (1,2)$ (a \emph{splitting pair}). We prove the other
direction whenever $q / p < \Phi$. In the Appendix, a variety of experimental
data is included, aiming to point out some directions for future work on GDWN
games.

4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga