56,004 research outputs found
Orientation reversal of manifolds
We call a closed, connected, orientable manifold in one of the categories
TOP, PL or DIFF chiral if it does not admit an orientation-reversing
automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly
chiral if it does not admit a self-map of degree -1. We prove that there are
strongly chiral, smooth manifolds in every oriented bordism class in every
dimension greater than two. We also produce simply-connected, strongly chiral
manifolds in every dimension greater than six. For every positive integer k, we
exhibit lens spaces with an orientation-reversing self-diffeomorphism of order
2^k but no self-map of degree -1 of smaller order.Comment: This is the update to the final version. 22 page
Diffusion in Networks and the Unexpected Virtue of Burstiness
Whether an idea, information, infection, or innovation diffuses throughout a
society depends not only on the structure of the network of interactions, but
also on the timing of those interactions. Recent studies have shown that
diffusion can fail on a network in which people are only active in "bursts",
active for a while and then silent for a while, but diffusion could succeed on
the same network if people were active in a more random Poisson manner. Those
studies generally consider models in which nodes are active according to the
same random timing process and then ask which timing is optimal. In reality,
people differ widely in their activity patterns -- some are bursty and others
are not. Here we show that, if people differ in their activity patterns, bursty
behavior does not always hurt the diffusion, and in fact having some (but not
all) of the population be bursty significantly helps diffusion. We prove that
maximizing diffusion requires heterogeneous activity patterns across agents,
and the overall maximizing pattern of agents' activity times does not involve
any Poisson behavior
Reversors and Symmetries for Polynomial Automorphisms of the Plane
We obtain normal forms for symmetric and for reversible polynomial
automorphisms (polynomial maps that have polynomial inverses) of the plane. Our
normal forms are based on the generalized \Henon normal form of Friedland and
Milnor. We restrict to the case that the symmetries and reversors are also
polynomial automorphisms. We show that each such reversor has finite-order, and
that for nontrivial, real maps, the reversor has order 2 or 4. The normal forms
are shown to be unique up to finitely many choices. We investigate some of the
dynamical consequences of reversibility, especially for the case that the
reversor is not an involution.Comment: laTeX with 5 figures. Added new sections dealing with symmetries and
an extensive discussion of the reversing symmetry group
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