3,602 research outputs found
Self-avoiding and plane-filling properties for terdragons and other triangular folding curves
We consider -folding triangular curves, or -folding t-curves, obtained
by folding times a strip of paper in , each time possibly left then
right or right then left, and unfolding it with angles. An example is
the well known terdragon curve. They are self-avoiding like -folding curves
obtained by folding times a strip of paper in two, each time possibly left
or right, and unfolding it with angles.
We also consider complete folding t-curves, which are the curves without
endpoint obtained as inductive limits of -folding t-curves. We show that
each of them can be extended into a unique covering of the plane by disjoint
such curves, and this covering satisfies the local isomorphism property
introduced to investigate aperiodic tiling systems. Two coverings are locally
isomorphic if and only if they are associated to the same sequence of foldings.
Each class of locally isomorphic coverings contains exactly
(resp. , or , ) isomorphism classes of coverings by
(resp. , , ) curves. These properties are partly similar to those
of complete folding curves.Comment: 15 pages, 3 figure
Peano-Gosper curves and the local isomorphism property
We consider unbounded curves without endpoints. Isomorphism is equivalence up
to translation. Self-avoiding plane-filling curves cannot be periodic, but they
can satisfy the local isomorphism property: We obtain a set of
coverings of the plane by sets of disjoint self-avoiding nonoriented curves,
generalizing the Peano-Gosper curves, such that:
1) each satisfies the local isomorphism property; any set of
curves locally isomorphic to belongs to ;
2) is the union of equivalence classes for the
relation " locally isomorphic to "; each of them contains
(resp. , , ) isomorphism classes of coverings by (resp.
, , ) curves.
Each gives exactly coverings by sets of oriented curves
which satisfy the local isomorphism property. They have opposite orientations.Comment: 15 pages, 5 figure
Decorated hypertrees
C. Jensen, J. McCammond and J. Meier have used weighted hypertrees to compute
the Euler characteristic of a subgroup of the automorphism group of a free
product. Weighted hypertrees also appear in the study of the homology of the
hypertree poset. We link them to decorated hypertrees after a general study on
decorated hypertrees, which we enumerate using box trees.---C. Jensen, J.
McCammond et J. Meier ont utilis\'e des hyperarbres pond\'er\'es pour calculer
la caract\'eristique d'Euler d'un sous-groupe du groupe des automorphismes d'un
produit libre. Un autre type d'hyperarbres pond\'er\'es appara\^it aussi dans
l'\'etude de l'homologie du poset des hyperarbres. Nous \'etudions les
hyperarbres d\'ecor\'es puis les comptons \`a l'aide de la notion d'arbre en
bo\^ite avant de les relier aux hyperarbres pond\'er\'es.Comment: nombre de pages : 3
Hypertree posets and hooked partitions
We adapt here the computation of characters on incidence Hopf algebras
introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded
posets. We then apply our results to the family of hypertree posets and
partition posets. As a consequence, we obtain some enumerative formulas and a
new proof for the computation of the Moebius numbers of the hypertree posets.
Moreover, we compute the coproduct of the incidence Hopf algebra and recover a
known formula for the number of hypertrees with fixed valency set and edge
sizes set.Comment: 18 page
Semi-pointed partition posets and Species
We define semi-pointed partition posets, which are a generalisation of
partition posets and show that they are Cohen-Macaulay. We then use multichains
to compute the dimension and the character for the action of the symmetric
groups on their homology. We finally study the associated incidence Hopf
algebra, which is similar to the Fa{\`a} di Bruno Hopf algebra.Comment: 27 page
Direct products and elementary equivalence of polycyclic-by-finite groups
We give an algebraic characterization of elementary equivalence for
polycyclic-by-finite groups. Using this characterization, we investigate the
relations between their elementary equivalence and the elementary equivalence
of the factors in their decompositions in direct products of indecomposable
groups. In particular we prove that the elementary equivalence of two such
groups G,H is equivalent to each of the following properties:
1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for
a strictly positive integer k;
2)AxG and AxH are elementarily equivalent for two elementarily equivalent
polycyclic-by-finite groups A,B.
It is not presently known if 1) implies elementary equivalence for any groups
G,H.Comment: 15 pages. Minor changes in pages 1 to 3, following the remarks of a
referee. The paper is presently publishe
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