4,153 research outputs found
A characterization of those automata that structurally generate finite groups
Antonenko and Russyev independently have shown that any Mealy automaton with
no cycles with exit--that is, where every cycle in the underlying directed
graph is a sink component--generates a fi- nite (semi)group, regardless of the
choice of the production functions. Antonenko has proved that this constitutes
a characterization in the non-invertible case and asked for the invertible
case, which is proved in this paper
The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable
We prove that a semigroup generated by a reversible two-state Mealy automaton
is either finite or free of rank 2. This fact leads to the decidability of
finiteness for groups generated by two-state or two-letter
invertible-reversible Mealy automata and to the decidability of freeness for
semigroups generated by two-state invertible-reversible Mealy automata
An explicit counterexample to the Lagarias-Wang finiteness conjecture
The joint spectral radius of a finite set of real matrices is
defined to be the maximum possible exponential rate of growth of long products
of matrices drawn from that set. A set of matrices is said to have the
\emph{finiteness property} if there exists a periodic product which achieves
this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that
every finite set of real matrices satisfies the finiteness
property. However, T. Bousch and J. Mairesse proved in 2002 that
counterexamples to the finiteness conjecture exist, showing in particular that
there exists a family of pairs of matrices which contains a
counterexample. Similar results were subsequently given by V.D. Blondel, J.
Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample
to the finiteness conjecture has so far been given. The purpose of this paper
is to resolve this issue by giving the first completely explicit description of
a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the
set \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\} we
give an explicit value of \alpha_* \simeq
0.749326546330367557943961948091344672091327370236064317358024...] such that
does not satisfy the finiteness property.Comment: 27 pages, 2 figure
Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture
Sofic and hyperlinear groups are the countable discrete groups that can be
approximated in a suitable sense by finite symmetric groups and groups of
unitary matrices. These notions turned out to be very deep and fruitful, and
stimulated in the last 15 years an impressive amount of research touching
several seemingly distant areas of mathematics including geometric group
theory, operator algebras, dynamical systems, graph theory, and more recently
even quantum information theory. Several longstanding conjectures that are
still open for arbitrary groups were settled in the case of sofic or
hyperlinear groups. These achievements aroused the interest of an increasing
number of researchers into some fundamental questions about the nature of these
approximation properties. Many of such problems are to this day still open such
as, outstandingly: Is there any countable discrete group that is not sofic or
hyperlinear? A similar pattern can be found in the study of II_1 factors. In
this case the famous conjecture due to Connes (commonly known as the Connes
embedding conjecture) that any II_1 factor can be approximated in a suitable
sense by matrix algebras inspired several breakthroughs in the understanding of
II_1 factors, and stands out today as one of the major open problems in the
field. The aim of these notes is to present in a uniform and accessible way
some cornerstone results in the study of sofic and hyperlinear groups and the
Connes embedding conjecture. The presentation is nonetheless self contained and
accessible to any student or researcher with a graduate level mathematical
background. An appendix by V. Pestov provides a pedagogically new introduction
to the concepts of ultrafilters, ultralimits, and ultraproducts for those
mathematicians who are not familiar with them, and aiming to make these
concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto
Formal study of plane Delaunay triangulation
This article presents the formal proof of correctness for a plane Delaunay
triangulation algorithm. It consists in repeating a sequence of edge flippings
from an initial triangulation until the Delaunay property is achieved. To
describe triangulations, we rely on a combinatorial hypermap specification
framework we have been developing for years. We embed hypermaps in the plane by
attaching coordinates to elements in a consistent way. We then describe what
are legal and illegal Delaunay edges and a flipping operation which we show
preserves hypermap, triangulation, and embedding invariants. To prove the
termination of the algorithm, we use a generic approach expressing that any
non-cyclic relation is well-founded when working on a finite set
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