656 research outputs found
Morphisms preserving the set of words coding three interval exchange
Any amicable pair \phi, \psi{} of Sturmian morphisms enables a construction
of a ternary morphism \eta{} which preserves the set of infinite words coding
3-interval exchange. We determine the number of amicable pairs with the same
incidence matrix in and we study incidence matrices associated
with the corresponding ternary morphisms \eta.Comment: 16 page
Derivated sequences of complementary symmetric Rote sequences
Complementary symmetric Rote sequences are binary sequences which have factor
complexity for all integers and whose
languages are closed under the exchange of letters. These sequences are
intimately linked to Sturmian sequences. Using this connection we investigate
the return words and the derivated sequences to the prefixes of any
complementary symmetric Rote sequence which is associated with a
standard Sturmian sequence . We show that any non-empty prefix of
has three return words. We prove that any derivated sequence of
is coding of three interval exchange transformation and we
determine the parameters of this transformation. We also prove that
is primitive substitutive if and only if is primitive
substitutive. Moreover, if the sequence is a fixed point of a
primitive morphism, then all derivated sequences of are also fixed
by primitive morphisms. In that case we provide an algorithm for finding these
fixing morphisms
Integers in number systems with positive and negative quadratic Pisot base
We consider numeration systems with base and , for quadratic
Pisot numbers and focus on comparing the combinatorial structure of the
sets and of numbers with integer expansion in base
, resp. . Our main result is the comparison of languages of
infinite words and coding the ordering of distances
between consecutive - and -integers. It turns out that for a
class of roots of , the languages coincide, while for other
quadratic Pisot numbers the language of can be identified only with
the language of a morphic image of . We also study the group
structure of -integers.Comment: 19 pages, 5 figure
Modular invariants and subfactors
In this lecture we explain the intimate relationship between modular
invariants in conformal field theory and braided subfactors in operator
algebras. Our analysis is based on an approach to modular invariants using
braided sector induction ("-induction") arising from the treatment of
conformal field theory in the Doplicher-Haag-Roberts framework. Many properties
of modular invariants which have so far been noticed empirically and considered
mysterious can be rigorously derived in a very general setting in the subfactor
context. For example, the connection between modular invariants and graphs (cf.
the A-D-E classification for ) finds a natural explanation and
interpretation. We try to give an overview on the current state of affairs
concerning the expected equivalence between the classifications of braided
subfactors and modular invariant two-dimensional conformal field theories.Comment: 25 pages, AMS LaTeX, epic, eepic, doc-class fic-1.cl
Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects
We consider three a priori totally different setups for Hopf algebras from
number theory, mathematical physics and algebraic topology. These are the Hopf
algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for
renormalization, and a Hopf algebra constructed by Baues to study double loop
spaces. We show that these examples can be successively unified by considering
simplicial objects, co-operads with multiplication and Feynman categories at
the ultimate level. These considerations open the door to new constructions and
reinterpretations of known constructions in a large common framework, which is
presented step-by-step with examples throughout. In this first part of two
papers, we concentrate on the simplicial and operadic aspects.Comment: This replacement is part I of the final version of the paper, which
has been split into two parts. The second part is available from the arXiv
under the title "Three Hopf algebras from number theory, physics & topology,
and their common background II: general categorical formulation"
arXiv:2001.0872
Three Hopf algebras and their common simplicial and categorical background
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common frameworkPreprin
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