1,231,794 research outputs found
Transport of patterns by Burge transpose
We take the first steps in developing a theory of transport of patterns from
Fishburn permutations to (modified) ascent sequences. Given a set of pattern
avoiding Fishburn permutations, we provide an explicit construction for the
basis of the corresponding set of modified ascent sequences. Our approach is in
fact more general and can transport patterns between permutations and
equivalence classes of so called Cayley permutations. This transport of
patterns relies on a simple operation we call the Burge transpose. It operates
on certain biwords called Burge words. Moreover, using mesh patterns on Cayley
permutations, we present an alternative view of the transport of patterns as a
Wilf-equivalence between subsets of Cayley permutations. We also highlight a
connection with primitive ascent sequences.Comment: 24 pages, 4 figure
Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks
Synchronization is of central importance in power distribution,
telecommunication, neuronal, and biological networks. Many networks are
observed to produce patterns of synchronized clusters, but it has been
difficult to predict these clusters or understand the conditions under which
they form, except for in the simplest of networks. In this article, we shed
light on the intimate connection between network symmetry and cluster
synchronization. We introduce general techniques that use network symmetries to
reveal the patterns of synchronized clusters and determine the conditions under
which they persist. The connection between symmetry and cluster synchronization
is experimentally explored using an electro-optic network. We experimentally
observe and theoretically predict a surprising phenomenon in which some
clusters lose synchrony while leaving others synchronized. The results could
guide the design of new power grid systems or lead to new understanding of the
dynamical behavior of networks ranging from neural to social
Simple marked mesh patterns
In this paper we begin the first systematic study of distributions of simple
marked mesh patterns. Mesh patterns were introduced recently by Br\"and\'en and
Claesson in connection with permutation statistics. We provide explicit
generating functions in several general cases, and develop recursions to
compute the numbers in question in some other cases. Certain -analogues are
discussed. Moreover, we consider two modifications of the notion of a marked
mesh pattern and provide enumerative results for them.Comment: 27 page
Contrast in Multipath Interference and Quantum Coherence
We develop a rigorous connection between statistical properties of an
interference pattern and the coherence properties of the underlying quantum
state. With explicit examples, we demonstrate that even for inaccurate
reconstructions of interference patterns properly defined statistical moments
permit a reliable characterization of quantum coherence.Comment: 10 page
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