88,622 research outputs found
A Geometric Approach to Combinatorial Fixed-Point Theorems
We develop a geometric framework that unifies several different combinatorial
fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing
them to be different geometric manifestations of the same topological
phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like
fixed-point theorems involving an exponential-sized label set; (2) a
generalization of Fan's parity proof of Tucker's Lemma to a much broader class
of label sets; and (3) direct proofs of several Sperner-like lemmas from
Tucker's lemma via explicit geometric embeddings, without the need for
topological fixed-point theorems. Our work naturally suggests several
interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
Local structure of abelian covers
We study normal finite abelian covers of smooth varieties. In particular we
establish combinatorial conditions so that a normal finite abelian cover of a
smooth variety is Gorenstein or locally complete intersection.Comment: Revised version; latex: 12 page
Roots, symmetries and conjugacy of pseudo-Anosov mapping classes
An algorithm is proposed that solves two decision problems for pseudo-Anosov
elements in the mapping class group of a surface with at least one marked fixed
point. The first problem is the root problem: decide if the element is a power
and in this case compute the roots. The second problem is the symmetry problem:
decide if the element commutes with a finite order element and in this case
compute this element. The structure theorem on which this algorithm is based
provides also a new solution to the conjugacy problem
The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics
This article is concerned with a general scheme on how to obtain constructive
proofs for combinatorial theorems that have topological proofs so far. To this
end the combinatorial concept of Tucker-property of a finite group is
introduced and its relation to the topological Borsuk-Ulam-property is
discussed. Applications of the Tucker-property in combinatorics are
demonstrated.Comment: 12 pages, 0 figure
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
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