Givental's non-linear Maslov index on lens spaces

Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications.Comment: 44 pages; v3: minor changes; v2: besides minor changes, we corrected a mistake in Corollary 1.3(iv

Families of short cycles on Riemannian surfaces

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a non-trivial element of the k'th homology group of the space of cycles and such that the mass of each cycle is bounded above by $C \max\{\sqrt{k}, \sqrt{g}\} \sqrt{Area(M)}$. This result is optimal up to a multiplicative constant.Comment: 16 pages, 3 figures. Exposition improved, to appear in Duke Mathematical Journa

Monotone Pieces Analysis for Qualitative Modeling

It is a crucial task to build qualitative models of industrial applications for model-based diagnosis. A Model Abstraction procedure is designed to automatically transform a quantitative model into qualitative model. If the data is monotone, the behavior can be easily abstracted using the corners of the bounding rectangle. Hence, many existing model abstraction approaches rely on monotonicity. But it is not a trivial problem to robustly detect monotone pieces from scattered data obtained by numerical simulation or experiments. This paper introduces an approach based on scale-dependent monotonicity: the notion that monotonicity can be defined relative to a scale. Real-valued functions defined on a finite set of reals e.g. simulation results, can be partitioned into quasi-monotone segments. The end points for the monotone segments are used as the initial set of landmarks for qualitative model abstraction. The qualitative model abstraction works as an iteratively refining process starting from the initial landmarks. The monotonicity analysis presented here can be used in constructing many other kinds of qualitative models; it is robust and computationally efficient