36,972 research outputs found
Finite subtraction games in more than one dimension
We study two-player impartial vector subtraction games (on tuples of
nonnegative integers) with finite rulesets, and solve all two-move games.
Through multiple computer visualizations of outcomes of two-dimensional
rulesets, we observe that they tend to partition the game board into periodic
mosaics on very few regions/segments, which can depend on the number of moves
in a ruleset. For example, we have found a five-move ruleset with an outcome
segmentation into six semi-infinite slices. We prove that games in two
dimensions are row/column eventually periodic. Several regularity conjectures
are provided. Through visualizations of some rulesets, we pose open problems on
the generic hardness of games in two dimensions.Comment: 38 page
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
Weighted Sobolev spaces and regularity for polyhedral domains
We prove a regularity result for the Poisson problem , u
|\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\
spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the
weight is given by the distance to the set of edges \cite{Babu70,
Kondratiev67}. In particular, we show that there is no loss of
\Kond{m}{a}--regularity for solutions of strongly elliptic systems with
smooth coefficients. We also establish a "trace theorem" for the restriction to
the boundary of the functions in \Kond{m}{a}(\PP)
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