25,188 research outputs found
The Number of Permutations Realized By a Shift
A permutation is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as . The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J. M. Amigó, S. Elizalde, and M. B. Kennel, J. Combin. Theory Ser. A, 115 (2008), pp. 485–504] that the shortest forbidden patterns of the shift on N symbols have length . In this paper we give a characterization of the set of permutations that are realized by the shift on N symbols, and we enumerate them according to their length
Forbidden patterns and shift systems
The scope of this paper is two-fold. First, to present to the researchers in
combinatorics an interesting implementation of permutations avoiding
generalized patterns in the framework of discrete-time dynamical systems.
Indeed, the orbits generated by piecewise monotone maps on one-dimensional
intervals have forbidden order patterns, i.e., order patterns that do not occur
in any orbit. The allowed patterns are then those patterns avoiding the
so-called forbidden root patterns and their shifted patterns. The second scope
is to study forbidden patterns in shift systems, which are universal models in
information theory, dynamical systems and stochastic processes. Due to its
simple structure, shift systems are accessible to a more detailed analysis and,
at the same time, exhibit all important properties of low-dimensional chaotic
dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a
dense set of periodic points), allowing to export the results to other
dynamical systems via order-isomorphisms.Comment: 21 pages, expanded Section 5 and corrected Propositions 3 and
Building GUTs from strings
We study in detail the structure of Grand Unified Theories derived as the
low-energy limit of orbifold four-dimensional strings. To this aim, new
techniques for building level-two symmetric orbifold theories are presented.
New classes of GUTs in the context of symmetric orbifolds are then constructed.
The method of permutation modding is further explored and SO(10) GUTs with both
or -plets are obtained. SU(5) models are also found through this
method. It is shown that, in the context of symmetric orbifold GUTs,
only a single GUT-Higgs, either a or a , can be present and it always
resides in an order-two untwisted sector. Very restrictive results also hold in
the case of . General properties and selection rules for string GUTs are
described. Some of these selection rules forbid the presence of some particular
GUT-Higgs couplings which are sometimes used in SUSY-GUT model building. Some
semi-realistic string GUT examples are presented and their properties briefly
discussed.Comment: 40 pages, no figures, Late
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