5,043 research outputs found
The absolute order on the hyperoctahedral group
The absolute order on the hyperoctahedral group is investigated. It is
proved that the order ideal of this poset generated by the Coxeter elements is
homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed.
Moreover, it is shown that every closed interval in the absolute order on
is shellable and an example of a non-Cohen-Macaulay interval in the absolute
order on is given. Finally, the closed intervals in the absolute order on
and which are lattices are characterized and some of their
important enumerative invariants are computed.Comment: 26 pages, 6 figures. Theorem 1.3 of the previous version of this
paper is omitted due to a gap in the proof
Defect tolerance: fundamental limits and examples
This paper addresses the problem of adding redundancy to a collection of physical objects so that the overall system is more robust to failures. In contrast to its information counterpart, which can exploit parity to protect multiple information symbols from a single erasure, physical redundancy can only be realized through duplication and substitution of objects. We propose a bipartite graph model for designing defect-tolerant systems, in which the defective objects are replaced by the judiciously connected redundant objects. The fundamental limits of this model are characterized under various asymptotic settings and both asymptotic and finite-size systems that approach these limits are constructed. Among other results, we show that the simple modular redundancy is in general suboptimal. As we develop, this combinatorial problem of defect tolerant system design has a natural interpretation as one of graph coloring, and the analysis is significantly different from that traditionally used in information redundancy for error-control codes.©201
Open sets satisfying systems of congruences
A famous result of Hausdorff states that a sphere with countably many points
removed can be partitioned into three pieces A,B,C such that A is congruent to
B (i.e., there is an isometry of the sphere which sends A to B), B is congruent
to C, and A is congruent to (B union C); this result was the precursor of the
Banach-Tarski paradox. Later, R. Robinson characterized the systems of
congruences like this which could be realized by partitions of the (entire)
sphere with rotations witnessing the congruences. The pieces involved were
nonmeasurable. In the present paper, we consider the problem of which systems
of congruences can be satisfied using open subsets of the sphere (or related
spaces); of course, these open sets cannot form a partition of the sphere, but
they can be required to cover "most of" the sphere in the sense that their
union is dense. Various versions of the problem arise, depending on whether one
uses all isometries of the sphere or restricts oneself to a free group of
rotations (the latter version generalizes to many other suitable spaces), or
whether one omits the requirement that the open sets have dense union, and so
on. While some cases of these problems are solved by simple geometrical
dissections, others involve complicated iterative constructions and/or results
from the theory of free groups. Many interesting questions remain open.Comment: 44 page
Digraphs and cycle polynomials for free-by-cyclic groups
Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be
represented by an expanding, irreducible train-track map. The automorphism
determines a free-by-cyclic group
and a homomorphism . By work of Neumann,
Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, has an open cone
neighborhood in whose integral points
correspond to other fibrations of whose associated outer automorphisms
are themselves representable by expanding irreducible train-track maps. In this
paper, we define an analog of McMullen's Teichm\"uller polynomial that computes
the dilatations of all outer automorphism in .Comment: 41 pages, 20 figure
Random generation of finitely generated subgroups of a free group
We give an efficient algorithm to randomly generate finitely generated
subgroups of a given size, in a finite rank free group. Here, the size of a
subgroup is the number of vertices of its representation by a reduced graph
such as can be obtained by the method of Stallings foldings. Our algorithm
randomly generates a subgroup of a given size n, according to the uniform
distribution over size n subgroups. In the process, we give estimates of the
number of size n subgroups, of the average rank of size n subgroups, and of the
proportion of such subgroups that have finite index. Our algorithm has average
case complexity \O(n) in the RAM model and \O(n^2\log^2n) in the bitcost
model
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