3,885 research outputs found
Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
We present a combinatorial approach to rigorously show the existence of fixed
points, periodic orbits, and symbolic dynamics in discrete-time dynamical
systems, as well as to find numerical approximations of such objects. Our
approach relies on the method of `correctly aligned windows'. We subdivide the
`windows' into cubical complexes, and we assign to the vertices of the cubes
labels determined by the dynamics. In this way we encode the dynamics
information into a combinatorial structure. We use a version of the Sperner
Lemma saying that if the labeling satisfies certain conditions, then there
exist fixed points/periodic orbits/orbits with prescribed itineraries. Our
arguments are elementary
The weighted hook length formula
Based on the ideas in [CKP], we introduce the weighted analogue of the
branching rule for the classical hook length formula, and give two proofs of
this result. The first proof is completely bijective, and in a special case
gives a new short combinatorial proof of the hook length formula. Our second
proof is probabilistic, generalizing the (usual) hook walk proof of
Green-Nijenhuis-Wilf, as well as the q-walk of Kerov. Further applications are
also presented.Comment: 14 pages, 4 figure
Minimum-Weight Edge Discriminator in Hypergraphs
In this paper we introduce the concept of minimum-weight edge-discriminators
in hypergraphs, and study its various properties. For a hypergraph , a function is said to be an {\it edge-discriminator} on if
, for all hyperedges , and
, for every two
distinct hyperedges . An {\it optimal
edge-discriminator} on , to be denoted by , is
an edge-discriminator on satisfying , where
the minimum is taken over all edge-discriminators on . We prove
that any hypergraph , with , satisfies ,
and equality holds if and only if the elements of are mutually
disjoint. For -uniform hypergraphs , it
follows from results on Sidon sequences that , and
the bound is attained up to a constant factor by the complete -uniform
hypergraph. Next, we construct optimal edge-discriminators for some special
hypergraphs, which include paths, cycles, and complete -partite hypergraphs.
Finally, we show that no optimal edge-discriminator on any hypergraph , with , satisfies
, which, in turn,
raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure
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