65,504 research outputs found
Pairwise Well-Formed Modes and Transformations
One of the most significant attitudinal shifts in the history of music
occurred in the Renaissance, when an emerging triadic consciousness moved
musicians towards a new scalar formation that placed major thirds on a par with
perfect fifths. In this paper we revisit the confrontation between the two
idealized scalar and modal conceptions, that of the ancient and medieval world
and that of the early modern world, associated especially with Zarlino. We do
this at an abstract level, in the language of algebraic combinatorics on words.
In scale theory the juxtaposition is between well-formed and pairwise
well-formed scales and modes, expressed in terms of Christoffel words or
standard words and their conjugates, and the special Sturmian morphisms that
generate them. Pairwise well-formed scales are encoded by words over a
three-letter alphabet, and in our generalization we introduce special positive
automorphisms of , the free group over three letters.Comment: 12 pages, 3 figures, paper presented at the MCM2017 at UNAM in Mexico
City on June 27, 2017, keywords: pairwise well-formed scales and modes,
well-formed scales and modes, well-formed words, Christoffel words, standard
words, central words, algebraic combinatorics on words, special Sturmian
morphism
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
Heisenberg-Weyl algebra revisited: Combinatorics of words and paths
The Heisenberg-Weyl algebra, which underlies virtually all physical
representations of Quantum Theory, is considered from the combinatorial point
of view. We provide a concrete model of the algebra in terms of paths on a
lattice with some decomposition rules. We also discuss the rook problem on the
associated Ferrers board; this is related to the calculus in the normally
ordered basis. From this starting point we explore a combinatorial underpinning
of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and
applications.Comment: 5 pages, 3 figure
Combinatorics on words in information security: Unavoidable regularities in the construction of multicollision attacks on iterated hash functions
Classically in combinatorics on words one studies unavoidable regularities
that appear in sufficiently long strings of symbols over a fixed size alphabet.
In this paper we take another viewpoint and focus on combinatorial properties
of long words in which the number of occurrences of any symbol is restritced by
a fixed constant. We then demonstrate the connection of these properties to
constructing multicollision attacks on so called generalized iterated hash
functions.Comment: In Proceedings WORDS 2011, arXiv:1108.341
A characterization of horizontal visibility graphs and combinatorics on words
An Horizontal Visibility Graph (for short, HVG) is defined in association
with an ordered set of non-negative reals. HVGs realize a methodology in the
analysis of time series, their degree distribution being a good discriminator
between randomness and chaos [B. Luque, et al., Phys. Rev. E 80 (2009),
046103]. We prove that a graph is an HVG if and only if outerplanar and has a
Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in
algebraic combinatorics [P. Flajolet and M. Noy, Discrete Math., 204 (1999)
203-229]. Our characterization of HVGs implies a linear time recognition
algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs
highlighting various connections with combinatorial statistics and introducing
the notion of a visible pair. With this technique we determine asymptotically
the average number of edges of HVGs.Comment: 6 page
New identities in dendriform algebras
Dendriform structures arise naturally in algebraic combinatorics (where they
allow, for example, the splitting of the shuffle product into two pieces) and
through Rota-Baxter algebra structures (the latter appear, among others, in
differential systems and in the renormalization process of pQFT). We prove new
combinatorial identities in dendriform dialgebras that appear to be strongly
related to classical phenomena, such as the combinatorics of Lyndon words,
rewriting rules in Lie algebras, or the fine structure of the
Malvenuto-Reutenauer algebra. One of these identities is an abstract
noncommutative, dendriform, generalization of the Bohnenblust-Spitzer identity
and of an identity involving iterated Chen integrals due to C.S. Lam.Comment: 16 pages, LaTeX. Concrete examples and applications adde
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