902 research outputs found
2-pile Nim with a Restricted Number of Move-size Imitations
We study a variation of the combinatorial game of 2-pile Nim. Move as in
2-pile Nim but with the following constraint:
Suppose the previous player has just removed say tokens from the
shorter pile (either pile in case they have the same height). If the next
player now removes tokens from the larger pile, then he imitates his
opponent. For a predetermined natural number , by the rules of the game,
neither player is allowed to imitate his opponent on more than
consecutive moves.
We prove that the strategy of this game resembles closely that of a variant
of Wythoff Nim--a variant with a blocking manoeuvre on diagonal
positions. In fact, we show a slightly more general result in which we have
relaxed the notion of what an imitation is.Comment: 18 pages, with an appendix by Peter Hegart
Impartial Games and Recursive Functions
Interest in 2-player impartial games often concerns the famous theory of Sprague-Grundy. In this thesis we study other aspects, bridging some gaps between combinatorial number theory, computer science and combinatorial games. The family of heap games is rewarding from the point of view of combinatorial number theory, partly because both the positions and the moves are represented simply by finite vectors of nonnegative integers. For example the famous game of Wythoff Nim on two heaps of tokens has a solution originating in Beatty sequences with modulus the Golden ratio. Sometimes generalizations of this game have similar properties, but mostly they are much harder to grasp fully. We study a spectrum of such variations, and our understanding of them ranges from being complete in the case of easier problems, to being very basic in the case of the harder ones. One of the most far reaching results concerns the convergence properties of a certain -operator for invariant subtraction games, introduced here to resolve an open problem in the area. The convergence holds for any game in any finite dimension. We also have a complete understanding of the reflexive properties of such games. Furthermore, interesting problems regarding computability can be formulated in this setting. In fact, we present two Turing complete families of impartial (heap) games. This implies that certain questions regarding their behavior are algorithmically undecidable, such as: Does a given finite sequence of move options alternate between N- and P-positions? Do two games have the same sets of P-positions? The notion of N- and P-positions is very central to the class of normal play impartial games. A position is in P if and only if it is safe to move there. This is virtually the only theory that we need. Therefore we hope that our material will inspire even advanced undergraduate students in future research projects. However we would not consider it impossible that the universality of our games will bridge even more gaps to other territories of mathematics and perhaps other sciences as well. In addition, some of our findings may apply as recreational games/mathematics
Restrictions of -Wythoff Nim and -complementary Beatty Sequences
Fix a positive integer . The game of \emph{-Wythoff Nim} (A.S.
Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner
the Queen'. Its set of -positions may be represented by a pair of increasing
sequences of non-negative integers. It is well-known that these sequences are
so-called \emph{complementary homogeneous}
\emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a
positive integer , we generalize the solution of -Wythoff Nim to a pair
of \emph{-complementary}---each positive integer occurs exactly
times---homogeneous Beatty sequences a = (a_n)_{n\in \M} and b = (b_n)_{n\in
\M}, which, for all , satisfies . By the latter property,
we show that and are unique among \emph{all} pairs of non-decreasing
-complementary sequences. We prove that such pairs can be partitioned into
pairs of complementary Beatty sequences. Our main results are that
\{\{a_n,b_n\}\mid n\in \M\} represents the solution to three new
'-restrictions' of -Wythoff Nim---of which one has a \emph{blocking
maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the
solution of Wythoff Nim satisfies the \emph{complementary equation}
. We generalize this formula to a certain '-complementary
equation' satisfied by our pair and . We also show that one may obtain
our new pair of sequences by three so-called \emph{Minimal EXclusive}
algorithms. We conclude with an Appendix by Aviezri Fraenkel.Comment: 22 pages, 2 figures, Games of No Chance 4, Appendix by Aviezri
Fraenke
Orientation matters for NIMreps
The problem of finding boundary states in CFT, often rephrased in terms of “NIMreps” of the fusion algebra, has a natural extension to CFT on non-orientable surfaces. This provides extra information that turns out to be quite useful to give the proper interpretation to a NIMrep. We illustrate this with several examples. This includes a rather detailed discussion of the interesting case of the simple current extension of A2 level 9, which is already known to have a rich structure. This structure can be disentangled completely using orientation information. In particular we find here and in other cases examples of diagonal modular invariants that do not admit a NIMrep, suggesting that there does not exist a corresponding CFT. We obtain the complete set of NIMreps (plus Moebius and Klein bottle coefficients) for many exceptional modular invariants of WZW models, and find an explanation for the occurrence of more than one NIMrep in certain cases. We also (re)consider the underlying formalism, emphasizing the distinction between oriented and unoriented string annulus amplitudes, and the origin of orientation-dependent degeneracy matrices in the latter.Fundação para a Ciência e Tecnologia, BD/13770/9
Graph kernels based on tree patterns for molecules
Motivated by chemical applications, we revisit and extend a family of
positive definite kernels for graphs based on the detection of common subtrees,
initially proposed by Ramon et al. (2003). We propose new kernels with a
parameter to control the complexity of the subtrees used as features to
represent the graphs. This parameter allows to smoothly interpolate between
classical graph kernels based on the count of common walks, on the one hand,
and kernels that emphasize the detection of large common subtrees, on the other
hand. We also propose two modular extensions to this formulation. The first
extension increases the number of subtrees that define the feature space, and
the second one removes noisy features from the graph representations. We
validate experimentally these new kernels on binary classification tasks
consisting in discriminating toxic and non-toxic molecules with support vector
machines
Hopf algebras and finite tensor categories in conformal field theory
In conformal field theory the understanding of correlation functions can be
divided into two distinct conceptual levels: The analytic properties of the
correlators endow the representation categories of the underlying chiral
symmetry algebras with additional structure, which in suitable cases is the one
of a finite tensor category. The problem of specifying the correlators can then
be encoded in algebraic structure internal to those categories. After reviewing
results for conformal field theories for which these representation categories
are semisimple, we explain what is known about representation categories of
chiral symmetry algebras that are not semisimple. We focus on generalizations
of the Verlinde formula, for which certain finite-dimensional complex Hopf
algebras are used as a tool, and on the structural importance of the presence
of a Hopf algebra internal to finite tensor categories.Comment: 46 pages, several figures. v2: missing text added after (4.5),
references added, and a few minor changes. v3: typos corrected, bibliography
update
Wythoff Wisdom
International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems
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