2,347 research outputs found
Permutation Complexity and the Letter Doubling Map
Given a countable set X (usually taken to be N or Z), an infinite permutation
of X is a linear ordering of X. This paper investigates the
combinatorial complexity of infinite permutations on N associated with the
image of uniformly recurrent aperiodic binary words under the letter doubling
map. An upper bound for the complexity is found for general words, and a
formula for the complexity is established for the Sturmian words and the
Thue-Morse word
Permutation Complexity Related to the Letter Doubling Map
Given a countable set X (usually taken to be the natural numbers or
integers), an infinite permutation, \pi, of X is a linear ordering of X. This
paper investigates the combinatorial complexity of infinite permutations on the
natural numbers associated with the image of uniformly recurrent aperiodic
binary words under the letter doubling map. An upper bound for the complexity
is found for general words, and a formula for the complexity is established for
the Sturmian words and the Thue-Morse word.Comment: In Proceedings WORDS 2011, arXiv:1108.341
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
Spin-Bits and N=4 SYM
We briefly review the spin-bit formalism, describing the non-planar dynamics
of the Super Yang-Mills SU(N) gauge theory. After
considering its foundations, we apply such a formalism to the sector of
purely scalar operators. In particular, we report an algorithmic formulation of
a deplanarizing procedure for local operators in the planar gauge theory, used
to obtain planarly-consistent, testable conjectures for the higher-loop
spin-bit Hamiltonians. Finally, we outlook some possible developments and
applications.Comment: 29 pages; contribution to the Proceedings of the 43rd Erice
International School of Subnuclear Physics ``Towards New Milestones in our
Quest to go Beyond the Standard Model'', Erice, Italy (29 August--7 September
2005); v2: some references adde
Morphisms, Symbolic sequences, and their Standard Forms
Morphisms are homomorphisms under the concatenation operation of the set of
words over a finite set. Changing the elements of the finite set does not
essentially change the morphism. We propose a way to select a unique
representing member out of all these morphisms. This has applications to the
classification of the shift dynamical systems generated by morphisms. In a
similar way, we propose the selection of a representing sequence out of the
class of symbolic sequences over an alphabet of fixed cardinality. Both methods
are useful for the storing of symbolic sequences in databases, like The On-Line
Encyclopedia of Integer Sequences. We illustrate our proposals with the
-symbol Fibonacci sequences
Canonical Representatives of Morphic Permutations
An infinite permutation can be defined as a linear ordering of the set of
natural numbers. In particular, an infinite permutation can be constructed with
an aperiodic infinite word over as the lexicographic order
of the shifts of the word. In this paper, we discuss the question if an
infinite permutation defined this way admits a canonical representative, that
is, can be defined by a sequence of numbers from [0, 1], such that the
frequency of its elements in any interval is equal to the length of that
interval. We show that a canonical representative exists if and only if the
word is uniquely ergodic, and that is why we use the term ergodic permutations.
We also discuss ways to construct the canonical representative of a permutation
defined by a morphic word and generalize the construction of Makarov, 2009, for
the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on
Words: 10th International Conference. arXiv admin note: text overlap with
arXiv:1503.0618
Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
We consider infinite sequences of superstable orbits (cascades) generated by
systematic substitutions of letters in the symbolic dynamics of one-dimensional
nonlinear systems in the logistic map universality class. We identify the
conditions under which the topological entropy of successive words converges as
a double exponential onto the accumulation point, and find the convergence
rates analytically for selected cascades. Numerical tests of the convergence of
the control parameter reveal a tendency to quantitatively universal
double-exponential convergence. Taking a specific physical example, we consider
cascades of stable orbits described by symbolic sequences with the symmetries
of quasilattices. We show that all quasilattices can be realised as stable
trajectories in nonlinear dynamical systems, extending previous results in
which two were identified.Comment: This version: updated figures and added discussion of generalised
time quasilattices. 17 pages, 4 figure
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