20,194 research outputs found
Universal Lyndon Words
A word over an alphabet is a Lyndon word if there exists an
order defined on for which is lexicographically smaller than all
of its conjugates (other than itself). We introduce and study \emph{universal
Lyndon words}, which are words over an -letter alphabet that have length
and such that all the conjugates are Lyndon words. We show that universal
Lyndon words exist for every and exhibit combinatorial and structural
properties of these words. We then define particular prefix codes, which we
call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in
bijection with the set of the shortest unrepeated prefixes of the conjugates of
a universal Lyndon word. This allows us to give an algorithm for constructing
all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201
The Rough Veronese variety
We study signature tensors of paths from a geometric viewpoint. The
signatures of a given class of paths parametrize an algebraic variety inside
the space of tensors, and these signature varieties provide both new tools to
investigate paths and new challenging questions about their behavior. This
paper focuses on signatures of rough paths. Their signature variety shows
surprising analogies with the Veronese variety, and our aim is to prove that
this so-called Rough Veronese is toric. The same holds for the universal
variety. Answering a question of Amendola, Friz and Sturmfels, we show that the
ideal of the universal variety does not need to be generated by quadrics
Algebras defined by Lyndon words and Artin-Schelter regularity
Let be a finite alphabet, and let be a
field. We study classes of graded -algebras , generated by and with a fixed set of obstructions
. Initially we do not impose restrictions on and investigate the case
when all algebras in have polynomial growth and finite
global dimension . Next we consider classes of
algebras whose sets of obstructions are antichains of Lyndon words. The
central question is "when a class contains Artin-Schelter
regular algebras?" Each class defines a Lyndon pair
which determines uniquely the global dimension, , and the
Gelfand-Kirillov dimension, , for every .
We find a combinatorial condition in terms of , so that the class
contains the enveloping algebra of a Lie
algebra . We introduce monomial Lie algebras defined by Lyndon
words, and prove results on Groebner-Shirshov bases of Lie ideals generated by
Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter
regular algebras of global dimensions and occurring as enveloping of standard monomial Lie algebras. The classification is made in
terms of their Lyndon pairs , each of which determines also the
explicit relations of .Comment: 49 page
The homotopy type of the loops on -connected -manifolds
For we compute the homotopy groups of -connected closed
manifolds of dimension . Away from the finite set of primes dividing
the order of the torsion subgroup in homology, the -local homotopy groups of
are determined by the rank of the free Abelian part of the homology.
Moreover, we show that these -local homotopy groups can be expressed as a
direct sum of -local homotopy groups of spheres. The integral homotopy type
of the loop space is also computed and shown to depend only on the rank of the
free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math.,
Birkhauser/Springer, 2018. arXiv admin note: text overlap with
arXiv:1510.0519
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