Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.Comment: Final version, 9 pages, 2 figure

Hyperbolic triangles without embedded eigenvalues

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.Comment: 65 pages, 4 figures, to appear in Annals of Mathematics http://annals.math.princeton.edu/articles/1159

Generic spectral simplicity of polygons

We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.Comment: length reduced to 6 pages, 1 figur

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.Comment: 41 page

International children's accelerometry database (ICAD): design and methods.

BACKGROUND: Over the past decade, accelerometers have increased in popularity as an objective measure of physical activity in free-living individuals. Evidence suggests that objective measures, rather than subjective tools such as questionnaires, are more likely to detect associations between physical activity and health in children. To date, a number of studies of children and adolescents across diverse cultures around the globe have collected accelerometer measures of physical activity accompanied by a broad range of predictor variables and associated health outcomes. The International Children's Accelerometry Database (ICAD) project pooled and reduced raw accelerometer data using standardized methods to create comparable outcome variables across studies. Such data pooling has the potential to improve our knowledge regarding the strength of relationships between physical activity and health. This manuscript describes the contributing studies, outlines the standardized methods used to process the accelerometer data and provides the initial questions which will be addressed using this novel data repository. METHODS: Between September 2008 and May 2010 46,131 raw Actigraph data files and accompanying anthropometric, demographic and health data collected on children (aged 3-18 years) were obtained from 20 studies worldwide and data was reduced using standardized analytical methods. RESULTS: When using ≥ 8, ≥ 10 and ≥ 12 hrs of wear per day as a criterion, 96%, 93.5% and 86.2% of the males, respectively, and 96.3%, 93.7% and 86% of the females, respectively, had at least one valid day of data. CONCLUSIONS: Pooling raw accelerometer data and accompanying phenotypic data from a number of studies has the potential to: a) increase statistical power due to a large sample size, b) create a more heterogeneous and potentially more representative sample, c) standardize and optimize the analytical methods used in the generation of outcome variables, and d) provide a means to study the causes of inter-study variability in physical activity. Methodological challenges include inflated variability in accelerometry measurements and the wide variation in tools and methods used to collect non-accelerometer data.RIGHTS : This article is licensed under the BioMed Central licence at http://www.biomedcentral.com/about/license which is similar to the 'Creative Commons Attribution Licence'. In brief you may : copy, distribute, and display the work; make derivative works; or make commercial use of the work - under the following conditions: the original author must be given credit; for any reuse or distribution, it must be made clear to others what the license terms of this work are

Euclidean Triangles Have No Hot Spots

We show that a second Neumann eigenfunction u of a Euclidean triangle has at most one (nonvertex) critical point p, and if p exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of u are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the `hot spots' conjecture for triangles in the plane.Comment: 29 pages, 3 figure