1,402 research outputs found
MV-algebras freely generated by finite Kleene algebras
If V and W are varieties of algebras such that any V-algebra A has a reduct
U(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) on
objects, and identically on homomorphisms. This functor U always has a left
adjoint F: W->V by general considerations. One calls F(B) the V-algebra freely
generated by the W-algebra B. Two problems arise naturally in this broad
setting. The description problem is to describe the structure of the V-algebra
F(B) as explicitly as possible in terms of the structure of the W-algebra B.
The recognition problem is to find conditions on the structure of a given
V-algebra A that are necessary and sufficient for the existence of a W-algebra
B such that F(B) is isomorphic to A. Building on and extending previous work on
MV-algebras freely generated by finite distributive lattices, in this paper we
provide solutions to the description and recognition problems in case V is the
variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely
generated--equivalently, finite. The proofs rely heavily on the Davey-Werner
natural duality for Kleene algebras, on the representation of finitely
presented MV-algebras by compact rational polyhedra, and on the theory of bases
of MV-algebras.Comment: 27 pages, 8 figures. Submitted to Algebra Universali
Topological signature for periodic motion recognition
In this paper, we present an algorithm that computes the topological
signature for a given periodic motion sequence. Such signature consists of a
vector obtained by persistent homology which captures the topological and
geometric changes of the object that models the motion. Two topological
signatures are compared simply by the angle between the corresponding vectors.
With respect to gait recognition, we have tested our method using only the
lowest fourth part of the body's silhouette. In this way, the impact of
variations in the upper part of the body, which are very frequent in real
scenarios, decreases considerably. We have also tested our method using other
periodic motions such as running or jumping. Finally, we formally prove that
our method is robust to small perturbations in the input data and does not
depend on the number of periods contained in the periodic motion sequence.Comment: arXiv admin note: substantial text overlap with arXiv:1707.0698
Good covers are algorithmically unrecognizable
A good cover in R^d is a collection of open contractible sets in R^d such
that the intersection of any subcollection is either contractible or empty.
Motivated by an analogy with convex sets, intersection patterns of good covers
were studied intensively. Our main result is that intersection patterns of good
covers are algorithmically unrecognizable.
More precisely, the intersection pattern of a good cover can be stored in a
simplicial complex called nerve which records which subfamilies of the good
cover intersect. A simplicial complex is topologically d-representable if it is
isomorphic to the nerve of a good cover in R^d. We prove that it is
algorithmically undecidable whether a given simplicial complex is topologically
d-representable for any fixed d \geq 5. The result remains also valid if we
replace good covers with acyclic covers or with covers by open d-balls.
As an auxiliary result we prove that if a simplicial complex is PL embeddable
into R^d, then it is topologically d-representable. We also supply this result
with showing that if a "sufficiently fine" subdivision of a k-dimensional
complex is d-representable and k \leq (2d-3)/3, then the complex is PL
embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in
version
Parametrized Complexity of Expansion Height
Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p
Topological Prismatoids and Small Simplicial Spheres of Large Diameter
We introduce topological prismatoids, a combinatorial abstraction of the
(geometric) prismatoids recently introduced by the second author to construct
counter-examples to the Hirsch conjecture. We show that the `strong -step
Theorem' that allows to construct such large-diameter polytopes from
`non--step' prismatoids still works at this combinatorial level. Then, using
metaheuristic methods on the flip graph, we construct four combinatorially
different non--step -dimensional topological prismatoids with
vertices. This implies the existence of -dimensional spheres with
vertices whose combinatorial diameter exceeds the Hirsch bound. These examples
are smaller that the previously known examples by Mani and Walkup in 1980 (
vertices, dimension ).
Our non-Hirsch spheres are shellable but we do not know whether they are
realizable as polytopes.Comment: 20 pages. Changes from v1 and v2: Reduced the part on shellability
and general improvement to accesibilit
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