2,560 research outputs found
Partitioning the power set of into -free parts
We show that for , in any partition of ,
the set of all subsets of , into parts, some
part must contain a triangle --- three different subsets
such that , , and have distinct representatives.
This is sharp, since by placing two complementary pairs of sets into each
partition class, we have a partition into triangle-free parts. We
also address a more general Ramsey-type problem: for a given graph , find
(estimate) , the smallest number of colors needed for a coloring of
, such that no color class contains a Berge- subhypergraph.
We give an upper bound for for any connected graph which is
asymptotically sharp (for fixed ) when , a cycle, path, or
star with edges. Additional bounds are given for and .Comment: 12 page
Phase locking of coupled lasers with many longitudinal modes
Detailed experimental and theoretical investigations on two coupled fiber
lasers, each with many longitudinal modes, reveal that the behavior of the
longitudinal modes depends on both the coupling strength as well as the
detuning between them. For low to moderate coupling strength only longitudinal
modes which are common for both lasers phase-lock while those that are not
common gradually disappear. For larger coupling strengths, the longitudinal
modes that are not common reappear and phase-lock. When the coupling strength
approaches unity the coupled lasers behave as a single long cavity with
correspondingly denser longitudinal modes. Finally, we show that the gradual
increase in phase-locking as a function of the coupling strength results from
competition between phase-locked and non phase-locked longitudinal modes.Comment: 3 pages, 4 figures, submitted to opt. let
Learning Design: reflections on a snapshot of the current landscape
The mounting wealth of open and readily available information and the swift evolution of social, mobile and creative technologies warrant a re-conceptualisation of the role of educators: from providers of knowledge to designers of learning. This need is being addressed by a growing trend of research in Learning Design. Responding to this trend, the Art and Science of Learning Design workshop brought together leading voices in the field and provided a forum for discussing its key issues. It focused on three thematic axes: practices and methods, tools and resources, and theoretical frameworks. This paper reviews some definitions of Learning Design and then summarises the main contributions to the workshop. Drawing upon these, we identify three key challenges for Learning Design that suggest directions for future research
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