158,389 research outputs found
A Combinatorial Analog of a Theorem of F.J.Dyson
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the
case n=2 was proposed by Tucker in 1945. Numerous generalizations and
applications of the Lemma have appeared since then. In 2006 Meunier proved the
Lemma in its full generality in his Ph.D. thesis. There are generalizations and
extensions of the Borsuk-Ulam theorem that do not yet have combinatorial
analogs. In this note, we give a combinatorial analog of a result of Freeman J.
Dyson and show that our result is equivalent to Dyson's theorem. As with
Tucker's Lemma, we hope that this will lead to generalizations and applications
and ultimately a combinatorial analog of Yang's theorem of which both
Borsuk-Ulam and Dyson are special cases.Comment: Original version: 7 pages, 2 figures. Revised version: 12 pages, 4
figures, revised proofs. Final revised version: 9 pages, 2 figures, revised
proof
A remark on Getzler's semi-classical approximation
Ezra Getzler notes in the proof of the main theorem of "The semi-classical
approximation for modular operads" that "A proof of the theorem could no doubt
be given using [a combinatorial interpretation in terms of a sum over
necklaces]; however, we prefer to derive it directly from Theorem 2.2". In this
note we give such a direct combinatorial proof using wreath product symmetric
functions.Comment: 7 page
Combinatorial Stokes formulas via minimal resolutions
We describe an explicit chain map from the standard resolution to the minimal
resolution for the finite cyclic group Z_k of order k. We then demonstrate how
such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn
implies "Dold's theorem" that there is no equivariant map from an n-connected
to an n-dimensional free Z_k-complex.
Thus we build a combinatorial access road to problems in combinatorics and
discrete geometry that have previously been treated with methods from
equivariant topology. The special case k=2 for this is classical; it involves
Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its
proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula
of Fan (1967), and Meunier's work (2006).Comment: 18 page
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
A functional combinatorial central limit theorem
The paper establishes a functional version of the Hoeffding combinatorial
central limit theorem. First, a pre-limiting Gaussian process approximation is
defined, and is shown to be at a distance of the order of the Lyapounov ratio
from the original random process. Distance is measured by comparison of
expectations of smooth functionals of the processes, and the argument is by way
of Stein's method. The pre-limiting process is then shown, under weak
conditions, to converge to a Gaussian limit process. The theorem is used to
describe the shape of random permutation tableaux.Comment: 23 page
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